The Dirichlet-type space
D
p
(
1
≤
p
≤
2
D^{p}\ (1 \leq p \leq 2
) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space
A
p
A^{p}
. Let
Φ
\Phi
be an analytic self-map of the disc and define
C
Φ
(
f
)
=
f
∘
Φ
C_{\Phi }(f) = f \circ \Phi
for
f
∈
D
p
f \in D^{p}
. The operator
C
Φ
:
D
p
→
D
p
C_{\Phi }: D^{p} \rightarrow D^{p}
is bounded (respectively, compact) if and only if a related measure
μ
p
\mu _{p}
is Carleson (respectively, compact Carleson). If
C
Φ
C_{\Phi }
is bounded (or compact) on
D
p
D^{p}
, then the same behavior holds on
D
q
(
1
≤
q
>
p
D^{q}\ (1 \leq q > p
) and on the weighted Dirichlet space
D
2
−
p
D_{2-p}
. Compactness on
D
p
D^{p}
implies that
C
Φ
C_{\Phi }
is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space
D
p
D^{p}
. Inner functions which induce bounded composition operators on
D
p
D^{p}
are discussed briefly.