  The class of support $\tau$-tilting modules was introduced to provide a
completion of the class of tilting modules from the point of view of mutations.
In this article we study $\tau$-tilting finite algebras, i.e. finite
dimensional algebras $A$ with finitely many isomorphism classes of
indecomposable $\tau$-rigid modules. We show that $A$ is $\tau$-tilting finite
if and only if very torsion class in $\mod A$ is functorially finite. We
observe that cones generated by $g$-vectors of indecomposable direct summands
of each support $\tau$-tilting module form a simplicial complex $\Delta(A)$. We
show that if $A$ is $\tau$-tilting finite, then $\Delta(A)$ is homeomorphic to
an $(n-1)$-dimensional sphere, and moreover the partial order on support
$\tau$-tilting modules can be recovered from the geometry of $\Delta(A)$.
Finally we give a bijection between indecomposable $\tau$-rigid $A$-modules and
bricks of $A$ satisfying a certain finiteness condition, which is automatic for
$\tau$-tilting finite algebras.
