Terrorists throwing hand grenades and firing automatic rifles today simultaneously attacked the check-in counters for the Israeli airline El Al at the international airports here and in Vienna. The attacks left 17 persons -- including three, and possibly four, Americans -- dead and 116 wounded, according to government and police officials.

Many of the casualties in Rome, including the Americans, were checking in for TWA Flight 841 to New York at a counter near El Al's when the terrorists struck. The three Americans killed were Natasha Simpson, 11, John Buonocore, 20, and Don Maland, 30.

In the shoot-out that followed the attack on the high-security section of the Leonardo da Vinci airport terminal on the outskirts of Rome, three of the four terrorists were killed in a hail of gunfire from Italian and Israeli security forces. A fourth terrorist, who told police he was a "Palestinian freedom fighter," was seriously wounded.

Although Italian police officials initially said five terrorists had staged the raid, tonight they corrected the figure to four.

In both attacks, the terrorists appeared to be Arabs, but their nationalities and political affiliations were not known.

When the shooting stopped, the airport was in chaos. Bodies were strewn across the floor, including that of a young terrorist in a yellow sweater, and the wounded were crawling and crying in pain. The windows on the front of the terminal were shot out and the driveway into the airport was full of wailing ambulances and police cars.

The attack, at about 9:15 this morning (3:15 EST), came as a similiar raid was staged by three terrorists on an El Al transit counter at Vienna's Schwechat International Airport, resulting in the death of one of three terrorists and two other persons and the wounding of 47 others.

It was not immediately clear what group staged the raids on the two busy airports packed with holiday travelers. But a man, speaking with a foreign accent, telephoned a Malaga, Spain, news service this afternoon saying the raids had been carried out by the organization of Abu Nidal, a renegade Palestinian who broke with Yasser Arafat, the chairman of the Palestine Liberation Organization in the early 1970s because he considered Arafat too moderate.

There was no confirmation of Abu Nidal's alleged involvement. In Washington, however, while U.S. officials sharply condemned the attack and declined to assign blame publicly, some said privately that initial evidence points to Abu Nidal's involvement. American and Egyptian officials also have said they suspect Abu Nidal was behind an alleged Libyan-backed hijacking of an Egyptair jetliner last month.

Israeli government officials blamed the PLO for the attacks and warned that it would retaliate harshly. The PLO office in Tunis issued a statement condemning the attacks and denying any responsibility.

According to initial reports tonight by Italian police officials, three, and possibly four, Americans were killed in the Rome attack, along with two Mexicans -- the military attache at the embassy here, Gen. Donato Merenda Acosta, and his secretary, who was seeing off him and his wife and children.

In addition, three Greeks, an Algerian, an Israeli who is believed to have been an El Al security guard and three terrorists, still not positively identified, were killed here.

Another man who died, and was found next to a briefcase with a U.S. passport inside, is thought to be a fourth American victim of the attack.

Among the 72 wounded at the Rome airport, according to U.S. Embassy sources, were another 12 Americans who were hospitalized for various injuries and seven who were treated and released.

In Vienna two travelers and one terrorist were killed and the two other terrorists were wounded and captured outside the airport after a police chase.

The three Americans whose deaths were confirmed were Buonocore, from Wilmington, Del.; Maland, from New Port Richey, Fla.; and Simpson, who was on her way to New York for home leave with her father, Victor, the news editor of the Associated Press bureau in Rome, her mother Daniella Petroff Simpson, also a journalist, and her 9-year-old brother, Michael.

The father and son were wounded in the attack.

The Simpson family, like many of the victims, was checking in at the nearby TWA counter.

According to police reconstructions and eyewitnesses' accounts, the four terrorists, who appeared to be Arabs, had entered the vast passenger terminal here among the thousands of passengers who pass through the airport each morning at its busiest hours.

The gunmen, according to the police, got to within about 50 yards of the El Al passenger check-in section, in what is the most tightly guarded area of the airport. The El Al area is overseen by machine gun-armed Italian carabinieri in bulletproof jackets who look down on the counter from a mezzanine while other police and security guards cruise the main terminal floor with guns at the ready.

The terrorists threw at least two hand grenades, according to witnesses and police officials, then opened up on the crowded area with Kalashnikov automatic assault rifles, spraying not only the El Al counter, but the neighboring TWA counter and a nearby coffee bar where other passengers were gathered.

Victor Simpson, of New York City, a 13-year resident of Rome, threw himself on his two children to try to protect them when the shooting began, according to Dennis Redmont, the AP bureau chief. Natasha was shot in the neck and the chest and died, according to Manuela Redmont, "in his arms." Michael Simpson suffered a severe stomach injury, while the children's father was injured in the right hand.

Simpson's wife escaped injury because she was walking her dog outside the terminal when the attack took place.

Shari Ruoff, a student from New York City who was waiting at the El Al counter to check in for a flight to Tel Aviv, said after being treated at a hospital for a leg wound, "I don't remember almost anything. One minute I was waiting to check in, the next moment I was thrown against the counter.

"There were people falling to the ground and others screaming. I felt a burning in my right leg and I collapsed to the floor."

According to police officials, the clash in the Rome airport lasted more than a minute.

The initial terrorist grenades were followed by a hail of gunfire from their Soviet-made Kalashnikov rifles.

Their attack was countered by pistol and machine-gun fire from the Italian police guarding the airport and plainclothed Israeli security men at the El Al counter, one of whom reportedly was killed.

Police recovered four Kalashnikovs from the terrorists and about 11 unexploded grenades, police sources said tonight.

The attack on the airport today capped a series of terrorist attacks in Italy this year that ranged from cafe bombings along Rome's Via Veneto to the October hijacking of the Italian cruise ship Achille Lauro as it sailed off the coast of Egypt.

The attack thus threatened to revive the heated Italian political debate over how to handle international terrorism and the country's policy toward the Middle East.

That debate in October, after the Achille Lauro hijacking, briefly brought down the five-party coalition government of Socialist Prime Minister Bettino Craxi, who has tried to steer Italy into a Mediterranean foreign policy seeking closer relations with the Arab world, including Arafat.

Defense Minister Giovanni Spadolini, the head of the Republican Party in the coalition and a man who has opposed Craxi's foreign policy in the region, resumed his criticism tonight, saying today's attacks "confirm the existence of a terrorist destabilization plan on an international level."

He called international terrorism "the gravest menace" to Italy's unity and stability as other voices in the government were raised concerning the ease with which Arabs, among others, are allowed to enter Italy without visas.

Interior Minister Oscar Luigi Scalfaro, who visited the airport immediately after the attack, defended security measures there, pointing out that today's attack, unlike previous ones within the transit area over the years, had been carried out in the main passenger entrance, where it was not easy to establish security cordons and personal screenings with metal detectors.

Police, meanwhile, said tonight that at least one of the terrorists in the attack was carrying a Moroccan passport of the same series that apparently was used by two of the three hijackers of an Egyptair passenger jet hijacked last month from Athens to Malta.

According to Italian news agency reports tonight, the wounded terrorist now under arrest was carrying a passport in the name of Tigriwi Abbegren. He is reported to be under tight guard in a ward of the Sant'Eugenio Hospital in a southern suburb of Rome.