The so-called Rankin-Selberg method is a powerful method, albeit more of an art than science, to establish analytic properties of L-functions. I will give an introduction mostly through examples. Time permitting, I will report on joint work with J. Cogdell and F. Shahidi developing a Rankin-Selberg integral for general spin groups, generalizing parts of results of D. Ginzburg and D. Soudry and Gelbart-Piatetski-Shapiro-Rallis for classical groups.