  We present a comprehensive theory of critical spaces for the broad class of
quasilinear parabolic evolution equations. The approach is based on maximal
$L_p$-regularity in time-weighted function spaces. It is shown that our notion
of critical spaces coincides with the concept of scaling invariant spaces in
case that the underlying partial differential equation enjoys a scaling
invariance. Applications to the vorticity equations for the Navier-Stokes
problem, convection-diffusion equations,the Nernst-Planck-Poisson equations in
electro-chemistry, chemotaxis equations, the MHD equations, and some other
well-known parabolic equations are given.
