In order to formalize the idea of the ``\(p\)-local structure'' of a finite group \(G\), the author introduced [in Bull. Soc. Math. Fr., Suppl., Mém. 47 (1976; Zbl 0355.20024)] the concept of the Frobenius category of \(G\) at the prime \(p\). This is the category \({\mathcal F}_G\) whose objects are the \(p\)-subgroups of \(G\), and the morphisms between two \(p\)-subgroups are those induced by inner automorphisms of \(G\) and by inclusions. The search for a more abstract concept which would allow a classification of \(p\)-local structure leads in this paper to the introduction of the Frobenius categories over a finite group \(P\). This work is motivated by results in finite group theory, block theory and homotopy theory. Most results of this paper are too technical to cite them here. We shall only mention the main notions which are introduced and studied. Denote by \({\mathcal F}_P\) the Frobenius category of a finite \(p\)-group \(P\). A \(P\)-category \(\mathcal F\) is a subcategory of the category of groups, having the same objects as \({\mathcal F}_P\), and all the homomorphisms are injective. Denoting by \({\mathcal F}(Q,R)\) the set of morphisms in \(\mathcal F\) from \(R\) to \(Q\), the category \(\mathcal F\) is called `divisible' if for any \(Q,R,T\leq P\), and \(\varphi\in{\mathcal F}(Q,R)\) and any group homomorphism \(\psi\colon T\to R\), \(\varphi\circ\psi\in{\mathcal F}(Q,T)\) if and only if \(\psi\in{\mathcal F}(R,T)\). Moreover, \(\mathcal F\) is a `Frobenius \(P\)-category' if in addition, \(\text{Int}(P)\) is a Sylow \(p\)-subgroup of \({\mathcal F}(P)\), and for any \(Q\leq P\), any \(K\leq\Aut(P)\) and any \(\varphi\in{\mathcal F}(P,Q)\) such that \(\varphi(Q)\) is fully \(^\varphi K\)-normalized in \(\mathcal F\), there are \(\varphi\in{\mathcal F}(P,Q\cdot N_P^K(Q))\) and \(\chi\in K\) such that \(\psi(u)=\varphi(\chi(u))\) for any \(u\in Q\), where \(N_P^K(Q)\) denotes the converse image of \(K\) in \(N_P(Q)\). Next, \(\mathcal F\)-selfcentralizing and \(\mathcal F\)-nilcentralized subgroups of \(P\) are introduced, and Frobenius \(P\)-categories are characterized in these terms. A generalization to Frobenius categories of the well-known Alperin's Fusion Theorem is given, in which \(\mathcal F\)-essential objects and the additive category \(\mathbb{Z}\mathcal F\) corresponding to \(\mathcal F\) are considered. Several constructions of Frobenius \(P\)-categories are presented, among them the hyperfocal Frobenius subcategory of \(\mathcal F\), which is the Frobenius category over the so called \(\mathcal F\)-hyperfocal subgroup of \(P\). This category is further studied in a subsequent paper of the author.