The purpose of this paper is to investigate the structure of the ring
D
(
R
)
D(R)
of all linear differential operators on the coordinate ring of an affine algebraic variety
X
X
(possibly reducible) over a field
k
k
(not necessarily algebraically closed) of characteristic zero, concentrating on the case that dim
X
⩽
1
X \leqslant 1
. In this case, it is proved that
D
(
R
)
D(R)
is a (left and right) noetherian ring with (left and right) Krull dimension equal to dim
X
X
, that the endomorphism ring of any simple (left or right)
D
(
R
)
D(R)
-module is finite dimensional over
k
k
, that
D
(
R
)
D(R)
has a unique smallest ideal
L
L
essential as a left or right ideal, and that
D
(
R
)
/
L
D(R)/L
is finite dimensional over
k
k
. The following ring-theoretic tool is developed for use in deriving the above results. Let
D
D
be a subalgebra of a left noetherian
k
k
-algebra
E
E
such that
E
E
is finitely generated as a left
D
D
-module and all simple left
E
E
-modules have finite dimensional endomorphism rings (over
k
k
), and assume that
D
D
contains a left ideal
I
I
of
E
E
such that
E
/
I
E/I
has finite length. Then it is proved that
D
D
is left noetherian and that the endomorphism ring of any simple left
D
D
-module is finite dimensional over
k
k
.