  Motivated by problems in which data are given over covering generating
families, we suggest a new cohomology theory for diffeological spaces, called
diffeological \v{C}ech cohomology, which is an exact $ \partial $-functor of
the section functor for sheaves on diffeological spaces. As applications, under
the situations of a setup, i) the generalized Mayer-Vietoris sequence for a
diffeological space is established; ii) a version of the de Rham theorem is
obtained, which connects diffeological \v{C}ech cohomology to the de Rham
cohomology. Moreover, we characterize the isomorphism classes of diffeological
fiber, principal, and vector bundles as (non-abelian) diffeological \v{C}ech
cohomology in degree 1.
