We study the Plateau Problem of finding an area minimizing
disk bounding a given Jordan curve in Alexandrov spaces with
curvature $≥ κ$. These are complete metric spaces with a lower
curvature bound given in terms of triangle comparison. Imposing
an additional condition that is satisfied by all Alexandrov spaces
according to a conjecture of Perel’man, we develop a harmonic
map theory from two dimensional domains into these spaces. In
particular, we show that the solution to the Dirichlet problem from
a disk is Hölder continuous in the interior and continuous up to
the boundary. Using this theory, we solve the Plateau Problem in
this setting generalizing classical results in Euclidean space (due
to J. Douglas and T. Rado) and in Riemannian manifolds (due to
C.B. Morrey).