In 1905, Minkowski proved that the discrete group SL(n,Z) is a lattice in the Lie group SL(n,R). This means that a fundamental domain has finite volume. More generally, by a theorem of Borel and Harish-Chandra, an arithmetic group in a simple Lie group is always a lattice. Conversely, by a celebrated theorem of Margulis, in a higher rank simple Lie group G any lattice is an arithmetic group. The aim of this lecture is to survey other arithmeticity criteria for discrete groups which are not assumed to be lattices. Generalizing work of Selberg and Hee Oh and solving with Miquel a conjecture of Margulis, we will see that a discrete subgroup in G is a non-cocompact arithmetic group if and only if it is Zariski dense and intersects cocompactly at least one horospherical subgroup U of G.