The VENONA project was a long-running and highly secret collaboration between the United States intelligence agencies and England's MI5. US Army codebreakers had intercepted large volumes of encrypted high-level Soviet diplomatic and intelligence traffic during and immediately after World War II. The British had stopped intercepting Soviet traffic, at Churchill's orders, shortly after Germany invaded the Soviet Union in 1941 and had no traffic to contibute to the project after that time. This traffic, some of which was thought to be encrypted with a one-time pad system, was stored and analyzed in relative secrecy by hundreds of cryptanalysts over a 40-year period starting in the early 1940s.
The British codename for VENONA was Bride. Some brilliant cryptanalysis by American and British codebreakers (the first steps were by a very young Meredith Gardner of what would become NSA) revealed that some of the one-time pad key material had incorrectly been reused by the Soviets, which allowed decryption (sometimes only partial) of some of the traffic. Over time, using assorted techniques ranging from traffic analysis to defector information, more of the messages were decrypted. Out of some hundreds of thousands of intercepted cyphertexts, it is claimed that under 3000 have been partially or wholly decrypted. Claims have made that information from physical theft of encryption pads (a partially burned one is reported to have been recovered by the Finns) to bugging embassy rooms in which text was entered into encrypting devices (and analyzing the keystrokes by listening to them being punched in), contributed to achieving as much plaintext as was recovered. These claims are less than fully supported in the open literature.
This decryption and cryptanalysis project became known to the Soviets not long after the first breaks. It is not clear whether the Soviets knew how much of the message traffic, or which messages, had been successfully decrypted. At least one Soviet agent, Kim Philby, was told about the project as part of his job as liaison between British and US intelligence. The project continued for decades, long after Philby left British intelligence.
The decrypted messages from Soviet aid missions, GRU spies, KGB spies, and some diplomatic traffic, known collectively as the VENONA papers, gave important insights into Soviet behavior in the period during which duplicate one-time pads were used. They also revealed the existence (and in some case the identities) of some American, Canadian, Australian, and British Soviet spies in research and in government (including Klaus Fuchs, Alan Nunn May (?), and at least one of the Cambridge Five spy ring (Donald Maclean).
After learning from their agent(s) of the US / British work, the Soviets naturally stopped reusing pads, after which their one-time pad traffic reverted to completely unreadable.