We give an introduction to open problems and their status concerning L^2-invariants. Besides the well-known Atiyah's Question and Singer's conjecture, we will discuss homological growth. We will discuss the Mahler measure of a polynomial and Lehmer's problem and explain their relevance for topology and number theory. We will pass to the more general notion of the Fuglede-Kadison determinant and discuss the extension of Lehmer's problem to arbitrary groups. Although we will state some theorems and partial answers, most of the talk is about work in progress.