After a slow beginning this year, Americans seem to be showing renewed interest in European travel. Spokesmen for the transatlantic travel industry report small but welcome increases in advance bookings this month.

Similarly, the State Department says applications for passports for foreign travel appear to be picking up. As of the beginning of June -- about when the application rate began to increase -- a total of 2,857,041 passports had been issued since Jan. 1, compared with 3,586,300 for the same period in 1985, a record year for U.S. travel to Europe.

The British Tourist Authority in New York is now getting up to 500 inquiries a day from prospective visitors to Great Britain -- a sharp jump over April and May when, says spokesman Bedford Pace, his office was "pretty quiet."

"The fact that the numbers are going up -- and not plummeting down -- is encouraging," says Trans World Airlines spokeswoman Sally McElwreath. "Frankly, the numbers are not sensational, but the downward spiral has stopped."

Pan Am spokesman Merle Richman agrees: "We certainly feel that things have bottomed out after a very bad April and May. The signs seem to be encouraging and positive as we move into late June and July and August." In May, only 39.7 percent of Pan Am's seats to Europe were filled, compared with 61.5 percent in May 1985.

In the past month, "our sales to Europe have quadrupled," says Marty Sitnick of the Travel Committee of Owings Mills, Md., a major operator of charter flights from Baltimore/Washington International Airport.

The travel industry realizes, of course, that this apparent trend could quickly be reversed if there are additional terrorist incidents.

The resurgence is being attributed to several factors. One is the bargain incentives, including free car rentals, hotel rooms and airline tickets, being offered by the airlines. "We all benefit from each other's advertising," says TWA's McElwreath. Other factors are the increased security precautions airlines and airports are adopting and human nature -- travelers tend to regain their confidence when there have been no recent alarms.

According to Pan Am's Richman, the upsurge in the airline's bookings seems to be mostly individual travelers, rather than people going with a group. "A greater percentage of people in groups are first- and second-time travelers," he says, "and they tend to be more skittish." Individual travelers also tend to be more flexible in their plans and can adopt a wait-and-see attitude, deciding to go to Europe -- or not go -- almost on the spur of the moment.

Air France reports its business traffic has continued to do well, and it is cautiously optimistic about improvements in vacation bookings this summer. "I know that our phone volume, our phone reservations are up," says spokesman Bruce Haxthausen. "That's a good sign."

About 60 percent of Air France's seats were filled in April compared with a load factor of 70 percent in April 1985. The airline has benefited from larger numbers of French travelers coming to the United States this year, because the decline in strength of the U.S. dollar has made this country more affordable to Europeans.

Here is a closer look at these and other developments in international travel:

Travel incentives: To attract passengers to Europe, a number of airlines are promoting special bargains.

For example, Pan Am is offering a free rental car with unlimited mileage if you and a companion book a trip to Europe before July 15 and travel before Dec. 15. When you return, you are eligible for the airline's "Winter in the Sun" offer. You can buy two tickets for the price of one to Pan Am's U.S. destinations as well as Bermuda and the Caribbean.

Northwest Orient has a similar program. If you fly to Europe, you get a week's free rental car and a two-for-one ticket offer to Northwest's U.S. destinations, including Hawaii and Alaska.

TWA's variation is a free rental car in Europe with 125 kilometers (85 miles) a day, plus free nights at participating Holiday Inns and Marriott Hotels. You stay from two to seven nights at one of the hotels and get the same number of nights free. There is also a two-for-one ticket offer to TWA's U.S. and Caribbean destinations.

The airlines also are offering sharply discounted youth fares for ages 12 to 24. Pan Am's round-trip youth fare between Washington and London is $396, compared with a regular 21-day advance purchase fare of $718. Reservations may be made no earlier than 72 hours of departure.

British Airways has taken a different approach. On June 10, it awarded 5,200 U.S. passengers free round trips to celebrate the 10th anniversary of Concorde service. Throughout the summer, it plans to offer daily in-flight prizes with a chance for special monthly drawings for such gifts as a new $100,000 Rolls-Royce, the use of a Concorde and crew for eight hours and a five-year lease on a London apartment.

Two budget airlines, Virgin Atlantic and People Express, are offering the lowest fares across the Atlantic -- they fly between Newark and London -- but the number of available seats is very limited. From Washington, you must also add the cost of getting to and from Newark. Virgin Atlantic has set a $178 round-trip fare (nonrefundable) for 30 seats on its daily flight. Travel must be completed by July 21. For flights after July 1, there is a 21-day advance-purchase requirement. Pending government approval, travelers who book no earlier than one day in advance can purchase a "late-saver" round-trip for $298. Virgin Atlantic's regular weekday round-trip fare is $578 and weekends, $598. For information: (212) 242-1330.

People Express, which has two flights daily, is offering round-trip fares on a limited basis on its 9:30 p.m. flight at $198, $298 and $398 (the cheapest fares are sold first). The fares are good through July 21. For flights after July 1, there is a 21-day advance-purchase requirement. The unrestricted round-trip fare is $498.

On the airline's 11:59 p.m. flight, seats are available at the discount rates of $198 and $298 through June 30. From July 1 to 21, the rates and advance-purchase requirement are the same as on the earlier flight.

Among other incentives, Inter-Continental Hotels is offering free weekends through Aug. 31 to guests who pay the regular rate for two nights during the week. The offer is good at 28 deluxe Inter-Continental and moderately priced Forum properties.

Inter-Continentals are located in Amsterdam, Athens, Cologne, Dusseldorf, Frankfurt, Geneva, Hamburg, Hanover, Lisbon, London, Luxembourg, Madrid, Paris and Vienna. Forums are located in Amsterdam, Belfast, Brussels, Budapest, Edinburgh, London, Munich, Rome and Wiesbaden. For information: (800) 327-0200.

A different kind of incentive, offered by the Travel Committee for its international charter flights, is an obligatory trip cancellation insurance policy that includes a terrorism provision. The cost for most passengers will be $12, which is added to the price of a tour or flight-only ticket.

The policy pays up to $750 if a trip must be cancelled because of illness, death of a family member, default of a tour operator or airline -- or if there is a terrorist incident within 30 days of a passenger's planned arrival in the city where the incident occurred. The insurance is provided by Access America, a subsidiary of Blue Cross and Blue Shield of the National Capital Area. For information: (301) 363-4900.

Health and safety: Both health and safety have become major concerns of transatlantic travelers as a result of terrorist attacks and the Soviet nuclear accident at Chernobyl.

As of this week, the only advisory issued by the State Department with regard to travel to Europe has been a warning to avoid the Soviet city of Kiev and the area within a 50- to 100-mile radius of the nuclear accident. Travelers to the Soviet Union and other East European countries also are advised to check with the U.S. embassy in each country about food or water restrictions, if any, resulting from nuclear contamination.

After the Chernobyl accident, the U.S. Public Heath Service got thousands of calls, many from prospective travelers, asking about radiation dangers in Europe, according to spokesman Ned Curran. The service currently is advising that travel in Europe, except in the vicinity of Chernobyl, "presents no hazard" due to radiation.

"All the information we have," says Curran, "is that radiation levels are nowhere near the danger level."

However, he adds, as a precaution for people still concerned about any possible contamination to their food, "We suggest it might not be advisable to drink fresh milk and dairy products, such as soft cheese" while in Europe. Other suggested protective measures, particularly in rural and out-of-the way locations, are to drink bottled water and "to stay away from raw fruit and vegetables."

Generally, he says, travelers can check with the U.S. embassy on local conditions and should abide by whatever restrictions, if any, a community has imposed. "It's a situation where everybody has to use their head. Nobody can give them a 100 percent guarantee."

The terrorist incidents of the past year have prompted U.S. overseas airlines, including Pan Am and TWA, to enhance their security measures as a means of bolstering passenger confidence.

Pan Am, for example, has set up what it calls an "elite special security unit" dubbed "Alert," aimed at providing "in-depth" and "highly visible " protection for passengers and crew. Alert will provide primary security at U.S. airports and backup security to government agencies at airports abroad.

Pan Am says its program will use "the most highly trained detection experts and the most sophisticated equipment to screen passengers, employes, baggage, airport facilities and aircraft." One immediate effect is that passengers are being asked to show up at the airport at least two hours before departure on an international flight.

Like Pan Am's, TWA's security measures emphasize visible efforts to screen passengers as well as a search of cabin, cockpit and baggage areas before passengers and baggage are loaded.

Pan Am, TWA and other U.S. airlines have added a surcharge of $5 each way on international flights to help offset the cost of security measures.

Meanwhile, the American Society of Travel Agents has published a new brochure, "Traveling Safely," which lists steps individual travelers can take to protect themselves while traveling abroad. Copies are available free from travel agents.

An alternative destination: Eastern Airlines, which earlier had planned to inaugurate a flight to Madrid this summer, decided instead to double its service to Buenos Aires. Eastern hopes to interest passengers who might have gone to Europe in trying a vacation in South America.

The airline has just introduced a Fly Away Buenos Aires package that begins at $860 (double occupancy) for a week and includes round-trip air fare from Miami, hotel accommodations, some meals and sightseeing.

"We've tried to put South America in a price category that's within reach," says spokeswoman Paula Musto. Once travelers arrive in South America, they will find the dollar is very strong and that there are good bargains in fine leather goods, sophisticated artwork and handicrafts. "Your money goes very far."

Customs surcharge: Beginning July 7, travelers entering the United States by air or sea from many destinations abroad will be charged a fee of $5. The per-person fee was authorized by Congress last year to help offset the cost of running the U.S. Customs Service.

The fee is applicable both to U.S. citizens and foreign visitors and will be collected by airlines and cruise companies. Exempt is travel from Mexico, Canada, the Caribbean and U.S. possessions abroad.

TRAVEL TALKS: Six American travel writers will offer advice on where to spend your vacation in a series of six lectures scheduled in July by the Smithsonian Resident Associate Program.

Entitled "Travel in America: New Tourist Destinations," the series opens July 10 with Eugene Fodor, founder of the Fodor guidebook firm, whose topic is "Traveling Our Great Land," a look at new trends in U.S. travel.

The lectures, 90 minutes each, will be held at 6 p.m. on Tuesdays and Thursdays from July 10 through July 29.

Other speakers: Norman Simpson, author of "Country Inns and Back Roads," on America's country inns; Nancy Shute, Washington correspondent for Outdoor magazine, on travel to Alaska, Hawaii and the Pacific Northwest; Margaret Zellers, author of guidebooks to the Caribbean, on Puerto Rico and the U.S. Virgin Islands; Judy Liberson, writer for Country magazine, on grand hotels and classic resorts for fall travel; and David Roberts, author of three books on mountaineering and exploration, on adventure travel. The fee for the series is $60 for Resident Associate members and $75 for nonmembers. For information and registration: 357-3030.

The lecture series is worth one college credit to students interested in enrolling in the two-year travel and tourism course offered by the Annandale campus of the Northern Virginia Community College. Students will interview the speakers and be required to prepare a research paper. The cost for the series is $30 for students plus a course fee of $17 for Virginia residents and $81 for out-of-state students. For information: 323-3474.