Astronomy and Cosmology are the disciplines that have developed techniques for determining the age of the universe. They have found it to be approximately 13 BILLION years old.
1. Distance of stars from the earth and the time it takes for their light to reach us.
2. Background radiation caused by the big bang.
Here are detailed explanations for the skeptical
How Astronomers Know the Vast Scale of Cosmic Time:
How we know the Age of the Cosmos
http://www.astrosociety.org/education/publications/tnl/56/index.html </a target="_blank">>
This special issue of an on-line newsletter for teachers sets
out to refute creationist claims of a young universe, and explains
how we know that the cosmos is 10-14 billion years old. (Produced
by the Astronomy Education Board of the American Astronomical
How we measure age of our planet
This useful page from the Talk.Origins site describes how we measure
the age of our planet and then dissects some of the common creationist
arguments for a younger Earth