Congruences, an essential aspect of number theory, need not only concern two numbers being congruent modulo, say a prime number p, but the idea extends to two objects being congruent modulo p. One aspect of an arithmetically interesting object is an L-function. For example, Wiles's celebrated proof of Fermat's last theorem may be construed as the L-function of an elliptic curve is in fact the L-function of a corresponding modular form. There is a principle in number theory that if two objects are congruent modulo a prime p, then the congruence can also be seen for the special values of L-functions attached to the objects. The context of modular forms provides lots of examples of congruences starting with the famous Ramanujan's 691-congruence: that the unique cusp form of weight 12 for the full modular group SL(2,Z) is congruent modulo 691 to an Eisenstein series also of weight 12 for SL(2,Z). One may then explore if and why this congruence persists for the special values of L-functions attached to these modular forms. In this talk, I will explain this circle of ideas using some very concrete examples of modular forms while invoking classical theorems of Shimura on the special values of Rankin-Selberg L-functions attached to these modular forms. I will describe the results, obtained in joint work with my student Narayanan, of certain computational experiments where one sees such congruences. Time-permitting, I will briefly sketch a framework involving the cohomology of arithmetic groups for GL(n)/Q which allows one to prove such congruences.