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.