  Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of
spaces using explicit cochain formulae for the cup-$i$ products; a family of
coherent homotopies derived from the broken symmetry of Alexander--Whitney's
chain approximation to the diagonal. He later defined his homonymous operations
for all primes using the homology of symmetric groups. This approach enhanced
the conceptual understanding of the operations and allowed for many advances,
but lacked the concreteness of their definition at the even prime. In recent
years, thanks to the development of new applications of cohomology, the need to
have an effectively computable definition of Steenrod operations has become a
key issue. Using the operadic viewpoint of May, this article provides such
definitions at all primes introducing multioperations that generalize the
Steenrod cup-$i$ products on the simplicial and cubical cochains of spaces.
