Spectral zeta functions is one type of generating function formed out of the spectrum of Laplace operators. In mathematical physics the interest in such functions started in the 1970s by papers by Dowker-Critchley and Hawking, for the purpose of defining the determinant of Laplacians. I will discuss these functions for some finite and infinite graphs, a topic that has not been much studied (in contrast to the Ihara zeta function, which is an entirely different function). In asymptotics ("thermodynamical limit") for families of torus graphs, classical number theoretic zeta functions appear. In these ways certain all-important problems in analytic number theory, such as the Riemann hypothesis, get surprising reinterpretations. Spectral zeta functions of graphs also relate to punctured surfaces, Verlinde formulas and certain hypergeometric functions.