  We give an introduction to (pseudo-)Finsler geometry and its connections. For
most results we provide short and self contained proofs. Our study of the
Berwald non-linear connection is framed into the theory of connections over
general fibered spaces pioneered by Mangiarotti, Modugno and other scholars.
The main identities for the linear Finsler connection are presented in the
general case, and then specialized to some notable cases like Berwald's,
Cartan's or Chern-Rund's. In this way it becomes easy to compare them and see
the advantages of one connection over the other. Since we introduce two
soldering forms we are able to characterize the notable Finsler connections in
terms of their torsion properties. As an application, the curvature symmetries
implied by the compatibility with a metric suggest that in Finslerian
generalizations of general relativity the mean Cartan torsion vanishes. This
observation allows us to obtain dynamical equations which imply a satisfactory
conservation law. The work ends with a discussion of yet another Finsler
connection which has some advantages over Cartan's and Chern-Rund's.
