  We introduce tropical vector bundles, morphisms and rational sections of
these bundles and define the pull-back of a tropical vector bundle and of a
rational section along a morphism. Afterwards we use the bounded rational
sections of a tropical vector bundle to define the Chern classes of this bundle
and prove some basic properties of Chern classes. Finally we give a complete
classification of all vector bundles on an elliptic curve up to isomorphisms.
