1982: US Places Cuba on State Sponsor List
US-Cuba relations have been sour since the 1959 Cuban communist revolution. The history of Cuba as a State Sponsor of terrorism didn't begin until 1982, though, when the State department placed the Caribbean island country on its list of those believed to be providing critical support to non-state terrorist groups.
Experts in the intelligence community generally don't think Cuba actively supports terrorist activity or that it poses any threat to national security. Most of the charges leveled against Cuba indicate that the US doesn't like Cuba's bad attitude toward US policy, but even the State Department can't provide convincing evident that Cuba actually acts to undermine US policy by supporting terrorism.
Cuba Doesn't Support the US led War on Terror
Cuba's main problem from the US point of view isn't what it does do to support terrorism, but what it doesn't do to stop it:
According to the State Department:
Cuba actively continued to oppose the U.S.-led Coalition prosecuting the global war on terror and has publicly condemned various U.S. policies and actions. To U.S. knowledge, Cuba did not attempt to track, block, or seize terrorist assets, although the authority to do so is contained in Cuba’s Law 93 Against Acts of Terrorism … No new counterterrorism laws were enacted, nor were any executive orders or regulations issued in this regard. To date, the Cuban Government has taken no action against al-Qaida or other terrorist groups. Cuba did not undertake any counterterrorism efforts in international and regional fora. Official government statements and the government-controlled press rarely speak out against al-Qaida or other designated Foreign Terrorist Organizations.
Cuba is friendly with other State Sponsors
The US also doesn't like the friends Cuba keeps, especially fellow State sponsor list members Iran and North Korea, with which Cuba has cooperation agreements in the areas of science and trade. The US maintains vague suspicions that Cuba's interest in biotechnology development, in combination with its science development agreements with Iran and North Korea, means it's up to no good.
The Cuban Government Refuses US Extradition Demands
According to the US, Cuba is harboring fugitives from American prosecution. These include murderers and hijackers, but no global terrorists of note. Cuba has agreed to stop providing safe haven to new US fugitives.
US charges that Cuba won't extradite US criminals has been weakened by the case of Luis Posada Carriles, who is a confirmed terrorist, and who has been given safe haven by the United States. Carriles, as some Americans have noted, was "our" terrorist—an anti-Castro Cuban who worked as a CIA operative in the past.