  The main goals of this paper are: i) To develop an abstract differential
calculus on metric measure spaces by investigating the duality relations
between differentials and gradients of Sobolev functions. This will be achieved
without calling into play any sort of analysis in charts, our assumptions
being: the metric space is complete and separable and the measure is Borel, non
negative and locally finite. ii) To employ these notions of calculus to
provide, via integration by parts, a general definition of distributional
Laplacian, thus giving a meaning to an expression like $\Delta g=\mu$, where
$g$ is a function and $\mu$ is a measure. iii) To show that on spaces with
Ricci curvature bounded from below and dimension bounded from above, the
Laplacian of the distance function is always a measure and that this measure
has the standard sharp comparison properties. This result requires an
additional assumption on the space, which reduces to strict convexity of the
norm in the case of smooth Finsler structures and is always satisfied on spaces
with linear Laplacian, a situation which is analyzed in detail.
