  Let $\Lambda$ be an Artin algebra and $K^b(proj(\Lambda))$ be the
triangulated category of bounded co-chain complexes in $proj(\Lambda).$ It is
well known that two-terms silting complexes in $K^b(proj(\Lambda))$ are
described by the $\tau$-tilting theory. The aim of this paper is to give a
characterization of certain $n$-term silting complexes in $K^b(proj(\Lambda))$
which are induced by $\Lambda$-modules. In order to do that, we introduce the
notions of $\tau_n$-rigid, $\tau_n$-tilting and $\tau_{n,m}$-tilting
$\Lambda$-modules. The latter is both a generalization of $\tau$-tilting and
tilting in $mod(\Lambda).$ It is also stated and proved some variant, for
$\tau_n$-tilting modules, of the well known Bazzoni's characterization for
tilting modules. We give some connections between $n$-terms presilting
complexes in $K^b(proj(\Lambda))$ and $\tau_n$-rigid $\Lambda$-modules.
Moreover, a characterization is given to know when a $\tau_n$-tilting
$\Lambda$-module is $n$-tilting. We also study more deeply the properties of
the $\tau_{n,m}$-tilting $\Lambda$-modules and their connections of being
$m$-tilting in some quotient algebras. We apply the developed
$\tau_{n,m}$-tilting theory to the finitistic dimension of $\Lambda.$ Finally,
at the end of the paper we discuss and state some open questions (conjectures)
that we consider crucial for the future develop of the $\tau_{n,m}$-tilting
theory.
