The theory of L-functions and zeta functions have been the key subject of mathematical research during the centuries since the Riemann zeta function was introduced and its important connection to the arithmetic of the integer was recognized. Though vast generalizations of the Riemann zeta function (for example, L-functions attached to motives, Galois representations, and automorphic representations) have been discovered and studied by many mathematicians, they are still mysterious analytic invariants. One approach to understand the origin of L-functions and their relation to number theory is studying p-adic L-functions and the Iwasawa main conjectures for a prime number p. In this talk, I will start from the simplest p-adic L-functions (due to Kubota-Leopoldt) and explain the idea of Iwasawa main conjecture, which gives a direct connection between p-adic L-functions and certain arithmetic objects called characteristic ideals. Hopefully we will be able to see how p-adic aspects of L-functions give some insight to their connection to arithmetic.