Let \(\mathbb{K}\) be a field and \(\mathbb{K}^* = G_ m\) the multiplicative group of \(\mathbb{K}\) viewed as an algebraic group over \(\mathbb{K}\). A \(\mathbb{K}\)-torus is an algebraic \(\mathbb{K}\)-group isomorphic to a direct product of such groups. Algebraic varieties which occur as closures of torus orbits for algebraic actions of a torus on an algebraic variety are called toric varieties. In this note we show how the structure of affine toric varieties \(M\), and those which occur by linear actions on projective spaces, can be analysed using the concept of affine algebraic monoids. The basic observation is that an affine toric variety carries a natural structure of an algebraic monoid. One of the main results states that the lattice of orbit closures of the torus \(T\) in \(M\) coincides with the finite lattice \(E(M)\) of idempotents of the monoid \(M\). To compute such a lattice in concrete situations one also needs a method to read it off the structural data given by the torus action. Such a method is provided by the identification of the set \(E(M)\) with a lattice of faces of a polyhedral cone in a real vector space which can be computed directly from the data defining the torus action. In the case of a linear action on a projective space the lattice of orbit closures can be identified with the lattice of faces of a convex polytope. In section I we collect some material on polyhedral cones in real vector spaces and in section II we consider finitely generated subsemigroups of \(\mathbb{Z}^ n\) and integral polyhedral cones. The main difficulty arising in this context is that such a finitely generated semigroup need not be saturated, i.e., the intersection of a cone in \(\mathbb{R}^ n\) with \(\mathbb{Z}^ n\). Section III contains some generalities on diagonalizable monoids over an arbitrary field \(\mathbb{K}\). In section IV we prove the structure theorem for toric monoids and we show how the lattice of idempotents of such monoids can be viewed as the lattice of faces of a polyhedral cone. -- These results are applied to the analysis of the lattice of orbit closures of a torus action on an affine variety and a linear action on projective space. Motivated by the observation that the structure of a toric monoid \(M\) is very similar to the structure of a compact abelian semigroup, we show in section V how toric varieties over local fields can also be described via compact semigroups.