Please note earlier start time.   I will introduce the concept of p-adic L-functions and describe some of their applications to questions originally considered by Kummer and his contemporaries, and later by Iwasawa. I will also try to illustrate the modern viewpoint by formulating the main conjecture of Iwasawa theory. If time permits, I will discuss generalizations of the main conjecture and recent applications of p-adic L-functions to the arithmetic of elliptic curves.