  We investigate stratified-algebraic vector bundles on a real algebraic
variety X. A stratification of X is a finite collection of pairwise disjoint,
Zariski locally closed subvarieties whose union is X. A topological vector
bundle on X is called a stratified-algebraic vector bundle if, roughly
speaking, there exists a stratification of X such that the restriction of the
bundle to each stratum is an algebraic vector bundle. In particular, every
algebraic vector bundle on X is stratified-algebraic. It turns out that
stratified-algebraic vector bundles have many surprising properties, which
distinguish them from algebraic and topological vector bundles.
