  We study the boundary regularity of solutions to the porous medium equation
$u_t = \Delta u^m$ in the degenerate range $m>1$. In particular, we show that
in cylinders the Dirichlet problem with positive continuous boundary data on
the parabolic boundary has a solution which attains the boundary values,
provided that the spatial domain satisfies the elliptic Wiener criterion. This
condition is known to be optimal, and it is a consequence of our main theorem
which establishes a barrier characterization of regular boundary points for
general -- not necessarily cylindrical -- domains in ${\bf R}^{n+1}$. One of
our fundamental tools is a new strict comparison principle between sub- and
superparabolic functions, which makes it essential for us to study both
nonstrict and strict Perron solutions to be able to develop a fruitful boundary
regularity theory. Several other comparison principles and pasting lemmas are
also obtained. In the process we obtain a rather complete picture of the
relation between sub/super\-para\-bolic functions and weak
sub/super\-solu\-tions.
