  We introduce the new concept of silting modules. These modules generalise
tilting modules over an arbitrary ring, as well as support $\tau$-tilting
modules over a finite dimensional algebra recently introduced by Adachi, Iyama
and Reiten. We show that silting modules generate torsion classes that provide
left approximations, and that every partial silting module admits an analogue
of the Bongartz complement. Furthermore, we prove that silting modules are in
bijection with 2-term silting complexes and with certain t-structures and
co-t-structures in the derived module category. We also see how some of these
bijections hold for silting complexes of arbitrary finite length.
