  The class of support $\tau$-tilting modules was introduced recently by
Adachi, Iyama and Reiten. These modules complete the class of tilting modules
from the point of view of mutations. Given a finite dimensional algebra $A$, we
study all basic support $\tau$-tilting $A$-modules which have given basic
$\tau$-rigid $A$-module as a direct summand. We show that there exist an
algebra $C$ such that there exists an order-preserving bijection between these
modules and all basic support $\tau$-tilting $C$-modules; we call this process
$\tau$-tilting reduction. An important step in this process is the formation of
$\tau$-perpendicular categories which are analogs of ordinary perpendicular
categories. Finally, we show that $\tau$-tilting reduction is compatible with
silting reduction and 2-Calabi-Yau reduction in appropiate triangulated
categories.
