  We define the full and reduced non-self-adjoint operator algebras associated
with \'etale categories and restriction semigroups, answering a question posed
by Kudryavtseva and Lawson in \cite{lawson}. Moreover, we define the
semicrossed product algebra of an \'etale action of a restriction semigroup on
a $C^*$-algebra, which turns out to be the key point when connecting the
operator algebra of a restriction semigroup with the operator algebra of its
associated \'etale category. We also prove that in the particular cases of
\'etale groupoids and inverse semigroups our operator algebras coincide with
the $C^*$-algebras of the referred objects.
