  We prove a general specialization theorem which implies stable irrationality
for a wide class of quadric surface bundles over rational surfaces. As an
application, we solve with the exception of two cases, the stable rationality
problem for any very general complex projective quadric surface bundle over the
projective plane, given by a symmetric matrix of homogeneous polynomials. Both
exceptions degenerate over a plane sextic curve and the corresponding double
cover is a K3 surface.
