During the Cold War the USA and USSR has massive intelligence and counterintelligence apparatuses set up to spy on each other. When the Cold War ended, how did these programs change? Obviously the KGB was disbanded, but was there still a massive focus on the other as the "enemy"? Were there captured spies that were traded, or known cases of spies or assets being left in country with no orders or contacts?
This is a big question, which invites an extremely lengthy answer, so I may need to split my response into multiple comments just to keep it coherent. Let's start by looking at Russia's post-Cold War intelligence community:
I'd contend that the KGB wasn't 'disbanded' in a true sense: legally, it was dissolved, and the Russian intelligence services did decline drastically in size, but the KGB was essentially dismantled into its component parts and reassembled into the post-Soviet security services — with a great deal of continuity in terms of personnel, tactics and techniques:
The First Chief Directorate — which was responsible for foreign intelligence, and which had always seen itself as an elite service somewhat separate from its sister departments — was separated from the KGB and reconstituted as the Foreign Intelligence Service (SVR). Freed from the byzantine structure of the KGB, the SVR quickly became bolder and more assertive in the internal politics of the post-Soviet Russian government than the FCD had been.
The Second Chief Directorate, responsible for counterintelligence, internal security and political policing, was first reorganised into the Ministry of Security, then morphed into the Federal Counterintelligence Service (FSK), and ultimately formed the nucleus of the Federal Security Service (FSB), established in 1995. The FSB has subsequently grown into the most powerful of the Russian state's security organs.
The Eighth Chief Directorate and Sixteenth Directorate, which handled SIGINT and COMINT, were merged into the Federal Agency of Government Communications and Information (FAPSI). In 2003, FAPSI was disbanded and its responsibilities handed over to the FSB.
The Ninth and Fifteenth Directorates, which provided the KGB's guards and security forces, were reformed into the Main Protection Administration (GUO) in 1992, eventually becoming the Federal Protective Service (FSO) in 1996.
The KGB Border Troops were reorganised into a new Federal Border Service (FPS), which was eventually placed under FSB control in 2003.
(As an aside, and at risk of running afoul of the 20-year rule: Vladimir Putin has taken a series of steps to re-centralise the Russian intelligence and security services. In 2003, he merged FAPSI and FPS into the FSB, and has turned the latter into an enormously powerful internal security apparatus. And last month, he announced plans to merge SVR, FSB and FSO into a new-but-rather-familiar Ministry of State Security.)
Russia's military intelligence service, the Main Intelligence Department (GRU) — which, though less well-known than the KGB, is generally believed to be both significantly larger and significantly more aggressive and expansive in its operations — survived the breakup of the Soviet Union essentially unchanged.
One consequence of this relatively ordered transition was that in many cases agents and networks already in place were simply inherited by the successor agencies (though not necessarily without some interruption.) Two of the KGB's most successful agents inside US intelligence — FBI counterintelligence agent Robert Hanssen and CIA counterintelligence officer Aldrich Ames — were both recruited by the KGB in the 1970s and 1980s and inherited by the SVR after 1991. They continued to provide information to Russia until their respective arrests in the mid-1990s.
After the breakup of the Soviet Union, the SVR continued to target the United States as a high priority: CIA intelligence officer Jim Nicholson, for example, was recruited by the Russians in around 1994. The ten Russian sleeper agents arrested by the FBI in 2010 were also placed in the US by the SVR, not the KGB.
The intelligence services are, like the KGB, one of the primary executors of Russian foreign policy, particularly under Putin. Russia was (and is) deeply concerned with political developments in its former empire, particularly as some of the former Soviet states liberalised and formed ever-closer relationships with Russia's traditional opponents in the west. Given Russia's long entanglement in Chechnya, its longstanding alliances in the Middle East, and the bitter memories of the Soviet invasion of Afghanistan, Russia has also been similarly concerned with the rising threat of Islamist terrorism since the 1990s.
The SVR and GRU have long been highly active in Europe, particularly in the former Soviet republics of Eastern Europe. Here's a good (pre-9/11) analysis of the SVR's position in the 1990s, written by a British observer in 2000. The FSB, meanwhile, assumes responsibility for operations in the former Soviet republics that now make up the Commonwealth of Independent States, and have been deeply involved in counterterrorism operations in restive regions like Chechnya and Abkhazia.
Andrei Soldatov and Irina Borogan's The New Nobility is a good read on the post-Soviet Russian security services. It traces the KGB's legacy in 1990s/2000s Russian politics and society, and the increasingly expansive and controversial role of the FSB. This paper from the European Council on Foreign Relations also does a good job of untangling the complex web of the Putin security and intelligence apparatuses. There are a couple of very good histories of Soviet intelligence which offer some good analysis of the issues which confronted the Soviet security services at the end of the Cold War: Christopher Andrew's The Sword and the Shield and Jonathan Haslam's Near and Distant Neighbours.
I'll try and circle back later to put together some thoughts on the post-Cold War evolution of the US intelligence community, to give you a sense of how things changed on the other side of the Iron Curtain.