  Our goal is to establish existence with suitable initial data of solutions to
general parabolic equation in one dimension, $u_t = L(u_x)_x$, where $L$ is
merely a monotone function. We also expose the basic properties of solutions,
concentrating on maximal possible regularity. Analysis of solutions with convex
initial data explains why we may call them {\it almost classical}. Some
qualitative aspects of solutions, like facets -- flat regions of solutions, are
studied too.
