  In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every
energy minimizing harmonic map from an Alexandrov space with curvature bounded
from below to an Alexandrov space with non-positive curvature is locally
H\"older continuous. In [39], F. H. Lin proposed a challenge problem: Can the
H\"older continuity be improved to Lipschitz continuity? J. Jost also asked a
similar problem about Lipschitz regularity of harmonic maps between singular
spaces (see Page 38 in [28]). The main theorem of this paper gives a complete
resolution to it.
