  In this paper we make a survey of some recent developments of the theory of
Sobolev spaces $W^{1,q}(X,\sfd,\mm)$, $1<q<\infty$, in metric measure spaces
$(X,\sfd,\mm)$. In the final part of the paper we provide a new proof of the
reflexivity of the Sobolev space based on $\Gamma$-convergence; this result
extends Cheeger's work because no Poincar\'e inequality is needed and the
measure-theoretic doubling property is weakened to the metric doubling property
of the support of $\mm$. We also discuss the lower semicontinuity of the slope
of Lipschitz functions and some open problems.

Analysis of PDEs