  Malcev described the congruences of the monoid $T_n$ of all full
transformations on a finite set $X_n=\{1, \dots,n\}$. Since then, congruences
have been characterized in various other monoids of (partial) transformations
on $X_n$, such as the symmetric inverse monoid $In_n$ of all injective partial
transformations, or the monoid $PT_n$ of all partial transformations.
  The first aim of this paper is to describe the congruences of the direct
products $Q_m\times P_n$, where $Q$ and $P$ belong to $\{T, PT,In\}$.
  Malcev also provided a similar description of the congruences on the
multiplicative monoid $F_n$ of all $n\times n$ matrices with entries in a field
$F$, our second aim is provide a description of the principal congruences of
$F_m \times F_n$.
  The paper finishes with some comments on the congruences of products of more
than two transformation semigroups, and a fairly large number of open problems.
