  In this article we will study semigroupoids, and more specifically inverse
semigroupoids. These are a common generalization to both inverse semigroups and
groupoids, and provide a natural language on which several types of dynamical
structures may be described. Moreover, this theory allows us to precisely
compare and simultaneously generalize aspects of both the theories of inverse
semigroups and groupoids. We begin by comparing and settling the differences
between two notions of semigroupoids which appear in the literature (one by
Tilson and another by Exel). We specialize this study to inverse semigroupoids,
and in particular an analogue of the Vagner-Preston Theorem is obtained. This
representation theorem leads to natural notions of actions, and more generally
$\land$-preactions and partial actions, of \'etale inverse semigroupoids, which
generalize topological actions of inverse semigroups and continuous actions of
\'etale groupoids. Many constructions which are commonplace in the theories of
inverse semigroups and groupoids are also generalized, and their categorical
properties made explicit. We finish this paper with a version of
non-commutative Stone duality for ample inverse semigroupoids, which utilizes
several of the aforementioned constructions.
