In this overview paper, presented at the meeting DANS14, Novi Sad, July3-7, 2014, we give some applications of Laplace transforms to analytic number theory. These include the classical circle and divisor problem, moments of |ζ( 1 2 + it)|, and a discussion of two functional equations connected to a work of Prof. Bogoljub Stanković.where I is an interval, and K 1 (s, t) is a suitable kernel function. If a problem involving the initial function f (t) can be solved by means of the transforms F (s), then by the inverse transformone can obtain information about f (t) itself. Here J is a suitable contour in C and K 2 (s, t) is another kernel function. Naturally the passage from (1.1) to (1.2) and back requires knowledge about the kernels, the convergence (existence) of the integrals that are involved, etc.