Let
G
G
be a finitely generated residually finite group and let
a
n
(
G
)
a_n(G)
denote the number of index
n
n
subgroups of
G
G
. If
a
n
(
G
)
≤
n
α
a_n(G) \le n^{\alpha }
for some
α
\alpha
and for all
n
n
, then
G
G
is said to have polynomial subgroup growth (PSG, for short). The degree of
G
G
is then defined by
deg
(
G
)
=
lim sup
log
a
n
(
G
)
log
n
\operatorname {deg}(G) = \limsup {{\log a_n(G)} \over {\log n}}
. Very little seems to be known about the relation between
deg
(
G
)
\operatorname {deg}(G)
and the algebraic structure of
G
G
. We derive a formula for computing the degree of certain metabelian groups, which serves as a main tool in this paper. Addressing a problem posed by Lubotzky, we also show that if
H
≤
G
H \le G
is a finite index subgroup, then
deg
(
G
)
≤
deg
(
H
)
+
1
\operatorname {deg}(G) \le \operatorname {deg}(H)+1
. A large part of the paper is devoted to the structure of groups of small degree. We show that
a
n
(
G
)
a_n(G)
is bounded above by a linear function of
n
n
if and only if
G
G
is virtually cyclic. We then determine all groups of degree less than
3
/
2
3/2
, and reveal some connections with plane crystallographic groups. It follows from our results that the degree of a finitely generated group cannot lie in the open interval
(
1
,
3
/
2
)
(1, 3/2)
. Our methods are largely number-theoretic, and density theorems à la Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.