Commutative associative multiplications on a space of functions can be defined in terms of multiplication kernels which are an infinite-dimensional analog of structure constants of multiplication in finite-dimensional case. Associativity constrain gives an integral equation for multiplication kernel. I will explain various ways of dealing with this integral equation in purely algebraic terms. In particular, connections with integrable systems will be discussed and a lot of examples will be constructed. The talk is based on the paper M. Kontsevich, A. Odesski Multiplication kernels, arXiv:2105.04238 [math.AG].