We usually do not think of new countries being formed, but some are not as old as you might think.
Bahrain - 15 August 1971 (from UK)
United Arab Emirates - 2 December 1971 (from UK)
Bangladesh -16 December 1971 (from West Pakistan)
Bahamas - 10 July 1973 (from UK)
Iran -1 April 1979 (Islamic Republic of Iran proclaimed)
Belize - 21 September 1981 (from UK)
Antigua and Barbuda - 1 November 1981 (from UK)
Brunei -1 January 1984 (from UK)
Marshall Islands - 21 October 1986 (from the US-administered UN trusteeship)
Federated States of Micronesia - 3 November 1986 (from the US-administered UN Trusteeship)
Lithuania - 11 March 1990 (from Soviet Union)
Namibia - 21 March 1990 (from South African mandate)
Armenia, Azerbaijan, Belarus, Estonia, Georgia, Kazakhstan, Latvia, Moldova, Russia, Turkmenistan, Ukraine, Uzbekistan - 1991 (from Soviet Union)
Croatia and Slovenia - 25 June 1991 (from Yugoslavia)
Macedonia -17 September 1991 (from Yugoslavia)
Bosnia and Herzegovina -1 March 1992 (from Yugoslavia)
Czech Republic and Slovakia -1 January 1993 (Czechoslovakia split)
Palau - 1 October 1994 (from the US-administered UN Trusteeship)
East Timor - 20 May 2002 (from Indonesia)
Serbia - May 2006 (after Montenegro declared independence from Yugoslavia)
Also, Cuba became independent from US on 20 May 1902.
Interesting to note that the Marshall Islands, Micronesia, and Palau are included and covered in the new US healthcare bill, even though they are independent countries and no longer under US Trusteeship.