  After summarizing some necessary preliminaries and tools, including Berwald
derivative and Lie derivative in pull-back formalism, we present ten equivalent
conditions, each of which characterizes Berwald manifolds among Finsler
manifolds. These range from Berwald's classical definition to the existence of
a torsion-free covariant derivative on the base manifold compatible with the
Finsler function and Aikou's characterization of Berwald manifolds. Finally, we
study some implications of V. Matveev's observation according to which
quadratic convexity may be omitted from the definition of a Berwald manifold.
These include, among others, a generalization of Z. I. Szab\'o's well-known
metrization theorem, and leads also to a natural generalization of Berwald
manifolds, to Berwald--Matveev manifolds.
