  We develop a mathematical theory for a class of compressible viscoelastic
rate-type fluids with stress diffusion. Our approach is based on the concepts
used in the nowadays standard theory of compressible Newtonian fluids as
renormalization, effective viscous flux identity, compensated compactness. The
presence of the extra stress, however, requires substantial modification of
these techniques, in particular, a new version of the effective viscous flux
identity is derived. With help of these tools, we show the existence of
global--in--time weak solutions for any finite energy initial data.
