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We define non-self-adjoint operator algebras associated with \'etale categories and restriction semigroups, answering a question posed by Ganna Kudryavtseva and Mark Lawson. We also show that these algebras have a reduced version when the category is left cancellative and the restriction semigroup is left-ample. 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., Comment: 56 pages