  The goal of this paper is to introduce a construction of a vector bundle on a
tropical variety. When the base is a tropical toric variety these tropicalize
toric vector bundles, and are described by the data of a valuated matroid and
some flats in the lattice of flats of the underlying matroid. The fibers are
tropical linear spaces. We define global sections for tropical toric vector
bundles, stability, and Jordan-H\"older and Harder-Narasimhan filtrations. Many
of these require additional modularity assumptions on the defining matroid.
