A classical result by Lovász asserts that two graphs G and H are isomorphic if and only if they have the same left profile, that is, for every graph F, the number of homomorphisms from F to G coincides with the number of homomorphisms from F to H. A similar result is also known to hold for right profiles, that is, two graphs G and H are isomorphic if and only if for every graph F, the number of homomorphisms from G to F coincides with the number of homomorphisms from H to F. During the past several years, there has been a study of equivalence relations that are relaxations of isomorphism obtained by restricting the left profile or the right profile to suitably restricted classes of graphs, instead of the class of all graphs. Furthermore, a notion of a query algorithm based on homomorphism counts was recently introduced and investigated. The aim of this talk is to present an overview of some of the main results in this area with emphasis on the differences between left homomorphism counts and right homomorphism counts.