  In this paper, motivated by a $\tau$-tilting version of the Brauer-Thrall
Conjectures, we study general properties of band modules and their
endomorphisms in the module category of a finite dimensional algebra. As an
application we describe properties of torsion classes containing band modules.
Furthermore, we show that a special biserial algebra is $\tau$-tilting finite
if and only if no band module is a brick. We also recover a criterion for the
$\tau$-tilting finiteness of Brauer graph algebras in terms of the Brauer
graph.
