Let
A
A
be a finite dimensional hereditary algebra over a field, with
n
n
simple
A
A
-modules. An
A
A
-module
T
A
T_A
with
n
n
pairwise nonisomorphic indecomposable direct summands and satisfying
Ex
t
1
(
T
A
,
T
A
)
=
0
{\text {Ex}}{{\text {t}}^1}({T_A},\,{T_A}) = 0
is called a tilting module, and its endomorphism ring
B
B
is a tilted algebra. A tilting module defines a (usually nonhereditary) torsion theory, and the indecomposable
B
B
-modules are in one-to-one correspondence to the indecomposable
A
A
-modules which are either torsion or torsionfree. One of the main results of the paper asserts that an algebra of finite representation type with an indecomposable sincere representation is a tilted algebra provided its Auslander-Reiten quiver has no oriented cycles. In fact, tilting modules are introduced and studied for any finite dimensional algebra, generalizing recent results of Brenner and Butler.