  Let $d,n$ be positive integers and $S$ be an arbitrary set of positive
integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in
S$. Consider the $S$-convolution of arithmetical functions given by (1.1),
where the sum is extended over the $S$-divisors of $n$.
  We determine the sets $S$ such that the $S$-convolution is associative and
preserves the multiplicativity of functions, respectively, and discuss other
basic properties of it. We give asymptotic formulae with error terms for the
functions $\sigma_S(n)$ and $\tau_S(n)$, representing the sum and the number of
$S$-divisors of $n$, respectively, for an arbitrary $S$. We improve the
remainder terms of these formulae and find the maximal orders of $\sigma_S(n)$
and $\tau_S(n)$ assuming additional properties of $S$. These results
generalize, unify and sharpen previous ones.
  We also pose some problems concerning these topics.
