In the talk 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. We explain the historical background, relations of the harmonicity to stochastic games and discuss some of the properties of strongly and weakly harmonic functions including Harnack estimates, maximum and comparison principles, the H\"older and the Lipshitz estimates and some differentiability properties. Moreover, we will discuss the mean value-harmonic functions in the setting of the Carnot--Carath\'eodory groups, focusing on regularity and relations of such functions to the sub-Laplace equation.

The talk is based on joint works with Gaczkowski, Gorka and Warhurst and related results by Soultanis and Kijowski.