  Consider the fractional powers $(A_{\operatorname{Dir}})^a$ and
$(A_{\operatorname{Neu}})^a$ of the Dirichlet and Neumann realizations of a
second-order strongly elliptic differential operator $A$ on a smooth bounded
subset $\Omega $ of ${\Bbb R}^n$. Recalling the results on complex powers and
complex interpolation of domains of elliptic boundary value problems by Seeley
in the 1970's, we demonstrate how they imply regularity properties in full
scales of $H^s_p$-Sobolev spaces and H\"older spaces, for the solutions of the
associated equations. Extensions to nonsmooth situations for low values of $s$
are derived by use of recent results on $H^\infty $-calculus. We also include
an overview of the various Dirichlet- and Neumann-type boundary problems
associated with the fractional Laplacian.
