  Given a continuous function on the boundary of a bounded open set in
$\mathbb{R}^d$ there exists a unique bounded harmonic function, called the
Perron solution, taking the prescribed boundary values at least at all regular
points (in the sense of Wiener) of the boundary. We extend this result to
vector-valued functions and consider several methods of constructing the Perron
solution which are classical in the real-valued case. We also apply our results
to solve elliptic and parabolic boundary value problems of vector-valued
functions.
