The Old Vic trustees said the theater's investigators had heard from 20 people who accused Spacey of a range of "inappropriate behaviors" during his 11-year run as artistic director of the renowned playhouse, from 2004 to 2015.
Spacey's transgressions are alleged to have ranged from "making people feel uncomfortable to sexually inappropriate behavior," the BBC reported.
Spacey is among the high-profile Hollywood figures caught up in a widening sexual harassment scandal, which began with revelations about film producer Harvey Weinstein.
Last month, actor Anthony Rapp accused Spacey of sexually assaulting him at a party when Rapp was 14 years old. Spacey responded that he did not remember the incident, but he did not deny it, offering Rapp his "sincerest apology for what would have been deeply inappropriate drunken behavior."
At the same time, Spacey used the incident to reveal that he is gay.
As other accusations emerged, Netflix suspended production of Spacey's popular "House of Cards" serial drama, and his performance in Ridley Scott's upcoming film "All the Money in the World" was stripped from the movie and reshot with actor Christopher Plummer.
The Old Vic hired London law firm Lewis Silkin to undertake an investigation and, promising anonymity, encouraged current and former employees at the theater to come forward. In all, 20 people have testified about incidents between 1995 and 2013.
In a lengthy statement about its investigation, the Old Vic cautioned: "No legal claims, formal grievances, formal disputes, settlement agreements or payments made or authorised were made at all in relation to Kevin Spacey during his tenure. It has also not been possible to verify any of these allegations, and it is important to note that Kevin Spacey has not commented on them."
Spacey was invited to participate in the investigation, but the theater said he did not respond.
The Guardian and BBC contacted Spacey's legal representatives for comment.
Spacey's successor as artistic director, Matthew Warchus, said: "These allegations have been a shock and a disturbing surprise to many of us. It is incorrect, unfair and irresponsible to say that everybody knew."
Before the allegations arose, Spacey's long run as artistic director at the Old Vic was roundly applauded. At first, London critics were suspicious, especially about whether the busy actor and global celebrity would have have the focus to turn the theatre around.
"What matters, in the end, is the work on stage and, after a patchy first season, Spacey has delivered the goods," wrote the Guardian's critic Michael Billington in 2015, remembering the years of Shakespeare beside American classics from Eugene O'Neill, Tennessee Williams and David Mamet.
Of the 20 alleged incidents, only one was reported during the time it occurred. The theater blamed fame.
"During Spacey's tenure, The Old Vic was in a unique position of having a Hollywood star at the helm around whom existed a cult of personality," the statement read. "The investigation found that his stardom and status at The Old Vic may have prevented people, and in particular junior staff or young actors, from feeling that they could speak up or raise a hand for help."
Staff who may have been aware of the alleged behavior "were unclear about how to respond; in some cases they did not consider allegations of misconduct to be serious and, where they clearly did, they said they did not feel confident that The Old Vic would take those allegations seriously given who he was," the report said.