Lattice point counting in Euclidean and hyperbolic spaces

I will give an overview of various problems that concern the asymptotic counting of lattice points (e.g. points with integer coordinates inside the Euclidean space) and the most standard techniques from analytic number theory used to tackle them. Often, one can find the leading terms in the required asymptotic formulas and there are open conjectures on the size of the error terms in these problems. In the hyperbolic setting, I will describe recent work with Chatzakos, Lester and Risager, to improve by a logarithmic factor the error term in the so-called hyperbolic circle problem.