  The first half of this work gives a survey of the fractional Laplacian (and
related operators), its restricted Dirichlet realization on a bounded domain,
and its nonhomogeneous local boundary conditions, as treated by
pseudodifferential methods. The second half takes up the associated heat
equation with homogeneous Dirichlet condition. Here we recall recently shown
sharp results on interior regularity and on $L_p$-estimates up to the boundary,
as well as recent H\"older estimates. This is supplied with new higher
regularity estimates in $L_2$-spaces using a technique of Lions and Magenes,
and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates
in the time-parameter) based on a general result of Amann. Moreover, it is
shown that an improvement to spatial $C^\infty $-regularity at the boundary is
not in general possible.
