  We present here a combinatorial method for computing cup-$i$ products and
Steenrod squares of a simplicial set $X$. This method is essentially based on
the determination of explicit formulae for the component morphisms of a higher
diagonal approximation (i.e., a family of morphisms measuring the lack of
commutativity of the cup product on the cochain level) in terms of face
operators of $X$. A generalization of this method to Steenrod reduced powers is
sketched. This description can be considered as a translation of the most
ancient definition of Steenrod squares to the general setting of the Simplicial
Topology.
