  We extend Gromov and Eliashberg-Mishachev's h-principle on manifolds to
stratified spaces. This is done in both the sheaf-theoretic framework of Gromov
and the smooth jets framework of Eliashberg-Mishachev. The generalization
involves developing 1) the notion of stratified continuous sheaves to extend
Gromov's theory, 2) the notion of smooth stratified bundles to extend
Eliashberg-Mishachev's theory. A new feature is the role played by homotopy
fiber sheaves. We show, in particular, that stratumwise flexibility of
stratified continuous sheaves along with flexibility of homotopy fiber sheaves
furnishes the parametric h-principle. We extend the Eliashberg-Mishachev
holonomic approximation theorem to stratified spaces. We also prove a
stratified analog of the Smale-Hirsch immersion theorem.
