What happened to the former colonial possessions of Europe?

The majority of the former colonial possessions of Europe became independent—most in the 1960s but some as late as the 1980s.
The United Kingdom had the biggest colonial empire. Former British colonies include: India, Pakistan, Kenya, Jamaica, Barbados, the Bahamas, and numerous other colonies.
France had the second largest empire. It lost its first acquisition, Haiti, with the uprising in 1793. Also in its possession were numerous West and North African countries, including Senegal, the Ivory Coast, Tunisia, and Morocco, as well as some Middle Eastern and Asian countries, including Lebanon and Vietnam, which, along with Laos, was a part of the former Indochina.
Economically, many colonial nations have struggled to gain footing. The purpose of colonialism was to exploit lands for resources. For example, the Caribbean islands were ideal places for growing sugar. Rubber was grown in Vietnam. Lands were razed to create plantations, which created an environmental impact. Part of the reason, for example, why Haiti is believed to be more vulnerable to inundation during natural disaster is both a lack of infrastructure and the long-term impact of deforestation.
Some former colonial possessions rejected Western democracy in the effort to redefine themselves. These include Cuba, which was first a Spanish possession, then briefly an American possession during the Age of Imperialism. It also includes Vietnam, whose fight over Communist rule resulted in a long war with the United States.
Other nations became vulnerable to corrupt rulers. This was particularly true in West African and Caribbean nations. Nigerian novelist Chinua Achebe's novel, A Man of the People, is based on the rise of corrupt national leaders in West Africa.
Today, former colonial nations work to address a host of challenges, which some are more equipped to address than others, including increased access to education, access to family planning and sex education, the development of infrastructure, sanitation, and how to avoid future exploitation. The latter is especially important, given China's current interests in Africa. It has been especially important to ensure a reciprocal relationship to avoid repeating past wrongs.
Some colonies are still a part of their "parent" countries. Martinique, Guadaloupe, and Guyana are still French territories. The United States has Guam and the Virgin Islands, and Puerto Rico has retained its commonwealth status.