Those affected said they “felt unable to raise concerns”, and the double Oscar winner “operated without sufficient accountability”.
The theatre said it “truly apologizes” for not creating a culture where people felt able to speak freely.
The Old Vic’s announcement follows recent allegations of harassment and predatory behavior made against Kevin Spacey while at the theatre.
Kevin Spacey has not responded to them.
His behavior was alleged to have ranged from making people feel uncomfortable to inappropriate behavior, the Old Vic investigation found.
No-one alleged rape but three people told the Old Vic they had contacted the police, while 14 of the 20 complainants were told by the investigators that they should consider going to the police.
There were 20 individual allegations and 16 were former staff, all of whom were men.
According to Lewis Silkin – the external law firm engaged by the theatre to conduct the investigation – more than half the allegations were said to have taken place inside the Old Vic.
The investigation also found there was not widespread knowledge of Kevin Spacey’s alleged behavior within the organization, the law firm said.
With the exception of one of the claims, none of the reported incidents were raised formally or informally with management.
Kevin Spacey was invited to participate in the investigation but the Old Vic said he did not respond.
The investigation is continuing so more people can come forward.
Current artistic director Matthew Warchus said he had “genuine and deep sympathy for all those who have come forward”.
He added: “Everyone is entitled to work in an environment free from harassment and intimidation.
“The Old Vic is now actively engaged in the process of healing and the process of prevention.”
Matthew Warchus described the allegations as “a shock and a disturbing surprise to many of us”.
“It is incorrect, unfair and irresponsible to say that everybody knew.
“But as a result of the investigation, what we have learnt is how better to call out this behavior in future.”
He said the findings would help “our industry as a whole, as together we rapidly evolve an intelligent new standard of protection and support in and around the workplace”.
The Old Vic’s executive director Kate Varah added that they did not want to “just rush out a statement”, saying: “We have not slept since this came out because we have been doing it in a robust, careful way.”
Kevin Spacey said on November 2 he was seeking treatment, in the wake of the allegations. However, there is no information about what kind of treatment the actor is seeking.
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