**Abstract:** The classification of topological states of matter is an important hot topic
in mathematical physics. In this talk I will describe a new approach to the
classification of topological quantum systems in class A-III which is based
on the study of a new category of vector bundles. The objects of this
category, the chiral vector bundles, are pairs constituted by a complex
vector bundle along with one of its automorphisms. We provide a
classification for the homotopy equivalence classes of these objects which
is based on the construction of a suitable classifying space. The
computation of the cohomology of the latter allows us to introduce a proper
set of characteristic cohomology classes: Some of those just reproduce the
ordinary Chern classes but there are also new odd-dimensional classes which
take care of the extra topological information introduced by the chiral
structure. Chiral vector bundles provide the proper geometric model for
topological quantum systems in class A-III, namely for systems endowed with
a (pseudo-)symmetry of chiral type. The classification of the chiral vector
bundles over sphere and tori (explicitly computable up to dimension 4)
recover the commonly accepted classification for topological insulators of
class A-III which is usually based on the K-group K1. However, this new
classification turns out to be even richer since it takes care also for the
possibility of non trivial Chern classes.