The aim of this paper is to find the global homological dimension of the ring of linear differential operators
R
[
θ
1
,
…
,
θ
u
]
R[{\theta _1}, \ldots ,{\theta _u}]
over a differential ring
R
R
with
u
u
commuting derivations. When
R
R
is a commutative noetherian ring with finite global dimension, the main theorem of this paper (Theorem 21) shows that the global dimension of
R
[
θ
1
,
…
,
θ
u
]
R[{\theta _1}, \ldots ,{\theta _u}]
is the maximum of
k
k
and
q
+
u
q + u
, where
q
q
is the supremum of the ranks of all maximal ideals
M
M
of
R
R
for which
R
/
M
R/M
has positive characteristic, and
k
k
is the supremum of the sums
r
a
n
k
(
P
)
+
d
i
f
f
d
i
m
(
P
)
rank(P) + diff\;dim(P)
for all prime ideals
P
P
of
R
R
such that
R
/
P
R/P
has characteristic zero. [The value
d
i
f
f
d
i
m
(
P
)
diff\;dim(P)
is an invariant measuring the differentiability of
P
P
in a manner defined in §3.] In case we are considering only a single derivation on
R
R
, this theorem leads to the result that the global dimension of
R
[
θ
]
R[\theta ]
is the supremum of gl
d
i
m
(
R
)
dim(R)
together with one plus the projective dimensions of the modules
R
/
J
R/J
, where
J
J
is any primary differential ideal of
R
R
. One application of these results derives the global dimension of the Weyl algebra in any degree over any commutative noetherian ring with finite global dimension.