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In this article we will study semigroupoids, and more specifically inversesemigroupoids. These are a common generalization to both inverse semigroups andgroupoids, and provide a natural language on which several types of dynamicalstructures may be described. Moreover, this theory allows us to preciselycompare and simultaneously generalize aspects of both the theories of inversesemigroups and groupoids. We begin by comparing and settling the differencesbetween two notions of semigroupoids which appear in the literature (one byTilson and another by Exel). We specialize this study to inverse semigroupoids,and in particular an analogue of the Vagner-Preston Theorem is obtained. Thisrepresentation theorem leads to natural notions of actions, and more generally$\land$-preactions and partial actions, of \'etale inverse semigroupoids, whichgeneralize topological actions of inverse semigroups and continuous actions of\'etale groupoids. Many constructions which are commonplace in the theories ofinverse semigroups and groupoids are also generalized, and their categoricalproperties made explicit. We finish this paper with a version ofnon-commutative Stone duality for ample inverse semigroupoids, which utilizesseveral of the aforementioned constructions.

\'Etale inverse semigroupoids - the fundamentals