ramanujan graphs are regular graphs whose nontrivial eigenvalues are small in absolute value. they have good expanding properties, and hence have broad applications in computer science. in particular, they give rise to very efficient communication networks. in this survey talk we shall discuss different explicit constructions of such graphs using group theory and number theory, and show how character sum estimates are used to estimate the eigenvalues. the eigenvalues of ramanujan graphs have close ties to fourier coefficients of automorphic forms. we shall give a flavor of this connection.