A polytope is defined as the convex hull of a finite number of points, or also as the bounded intersection of a finite number of half-spaces. Section 7.1 recalls the equivalence of these definitions, and gives the definition of the faces of a polytope. Polarity is also introduced in this section. The polarity centered at O is a dual transform between points and hyperplanes in Euclidean spaces which induces a duality on the set of polytopes containing the center O. Simple and simplicial polytopes are also defined in this section. Section 7.2 takes a close interest in the combinatorics of polytopes. It contains a proof of Euler's relation and the Dehn–Sommerville relations. Euler's relation is the only linear relation between the numbers of faces of each dimension of any polytope, and the Dehn–Sommerville relations are linear relations satisfied by simple polytopes. These relations can be used to show the celebrated upper bound theorem which bounds the number of faces of all dimensions of a d-dimensional polytope as a function of the number of its vertices, or facets. Considering cyclic polytopes shows that the upper bound theorem yields the best possible asymptotic bound. Linear unbounded convex sets enjoy similar properties and are frequently encountered in the rest of this book. Section 7.3 extends these definitions and properties to unbounded polytopes. A simple method to enforce coherence in these definitions is to consider the Euclidean space as embedded in the oriented projective space, an oriented version of the classical projective space.