  In this paper we study the (nonlocal) torsional rigidity in the ambient space
of random walk spaces. We get the relation of the (nonlocal) torsional rigidity
of a set $\Omega$ with the spectral $m$-heat content of $\Omega$, what gives
rise to a complete description of the nonlocal torsional rigidity of $\Omega$
by using uniquely probability terms involving the set $\Omega$; and recover the
first eigenvalue of the nonlocal Laplacian with homogeneous Dirichlet boundary
conditions by a limit formula using these probability term. For the random walk
in $\R^N$ associated with a non singular kernel, we get a nonlocal version of
the Saint-Venant inequality, and, under rescaling we recover the classical
Saint-Venant inequality. We study the nonlocal $p$-torsional rigidity and its
relation with the nonlocal Cheeger constants. We also get a nonlocal version of
the P\'{o}lya-Makai-type inequalities. We relate the torsional rigidity given
here for weighted graphs with the torsional rigidity on metric graphs.
