  We introduce and study strongly and weakly harmonic functions on metric
measure spaces defined via the mean value property holding for all and,
respectively, for some radii of balls at every point of the underlying domain.
Among properties of such functions we investigate various types of Harnack
estimates on balls and compact sets, weak and strong maximum principles,
comparison principles, the H\"older and the Lipshitz estimates and some
differentiability properties. The latter one is based on the notion of a weak
upper gradient. The Dirichlet problem for functions satisfying the mean value
property is studied via the dynamical programming method related to stochastic
games. We employ the Perron method to construct a harmonic function with
continuous boundary data. Finally, we discuss and prove the Liouville type
theorems.
  Our results are obtained for various types of measures: continuous with
respect to a metric, doubling, uniform, measures satisfying the annular decay
condition. Relations between such measures are presented as well. The
presentation is illustrated by examples.
