A description is given of those sequences
S
=
(
S
(
0
)
,
S
(
1
)
,
…
,
S
(
l
)
)
\mathbf {S}= (S(0),S(1),\dots ,S(l))
of simple modules over a finite dimensional algebra for which there are only finitely many uniserial modules with consecutive composition factors
S
(
0
)
,
…
,
S
(
l
)
S(0),\dots , S(l)
. Necessary and sufficient conditions for an algebra to permit only a finite number of isomorphism types of uniserial modules are derived. The main tools in this investigation are the affine algebraic varieties parametrizing the uniserial modules with composition series
S
\mathbf {S}
.