Many people left Cuba after the Revolution. They were mainly members of the middle and upper classes who were being heavily taxed by the government. Almost all of the extremely wealthy members of society fled, as they lost much of their property to the new government. Others who had sympathized with Batista feared for their safety.
Nearly everyone who left Cuba came to the United States, with many settling in Miami and other cities in Florida. Ever since, they have tried to overthrow Fidel Castro, so that they can return to Cuba and recover their lost fortunes.
The United States Government, as an enemy of the Castro Government as well, had supported them in their early activities.
These days, the political importance of Florida, as well as the sway of large campaign contributions, have given many anti-Castro groups great political power in the United States. Although their activities have become less direct, many still actively work for Fidel Castro's overthrow.
In the years after the Revolution, many anti-Castro exiles worked with the CIA or alone to bring about their return to power. The best known event occurred at the Bay of Pigs.
Cuban-Americans lead by the CIA and directly commanded by Batista's officers began training to invade Cuba. After an air raid on the island, the landed their forces at two points. at two places, being thoroughly defeated. They had expected support through a general uprising by Cubans still in Cuba, but they never got.
After that, groups such as Omega 7 and Alpha 66 came into existence. These terrorist organizations have bombed buildings, airplanes and factories in Cuba, and have even carried out assassinations around the world.
Many foreign embassies have been bombed after countries acted against the anti-Castro cause. Even prominent anti-Castro Cuban-Americans who don't support exactly the political agendas of these organizations have been assassinated. Thousands have been killed, and they have not been able to retake Cuba.
Today, there is less violence. The anti-Castro groups, such as the Cuban American National Foundation, today concentrate on influencing politics to bring change to Cuba. Most are extremely right wing and probably not be willing to pay for the continued existence of the social progress made by the Revolution, should they take power.
Most people living in Cuba don't like the prospect of the exiles' return. None of them believe that these wealthy citizens will continue to provide them with free education, health care, housing and other essential needs. These groups, which say they want to bring democracy to Cuba, would not stand much of a chance of winning an election.