A zonotope Z with n zones in Euclidean space \({\mathbb{R}}^ d\) (1\(\leq d\leq n)\) is the Minkowski sum of n line segments. Using the fact that Z can be decomposed into \(\left( \begin{matrix} n\\ d\end{matrix} \right)\) parallelotopes, a formula for the volume V(Z) of Z is established (Section 1). It is used to investigate the problem of maximizing the volume of a zonotope Z under various constraints - for example, that (1) the sum of the squares of the lengths of the generating line segments of Z be constant \((=d)\), or that (2) Z be the orthogonal projection of a unit n-cube; these two constraints (1),(2) lead to the same maximal figures (Section 2)! Section 3 presents a new proof of a necessary condition for the problem of maximizing the volume of the orthogonal projection of a convex polytope into a hyperplane; this leads to a necessary condition for a zonotope to attain the largest volume in the families (1) and (2), as well as in the family of zonotopes whose generating line segments are of equal length. Finally, various upper bounds for V(Z) and other specific results concerning the zonotopes Z of type (1) and (2) are given; for instance, the triacontahedron has the largest volume among all orthogonal projections of a 6-cube into \(R^ 3\).