If ϑ is a non-constant analytic function defined on the unit disk D such that ϑ( D ) ⊂ D , the composition operator C ϑ is the operator on the Hardy space H 2 of the unit disk defined by C ϑ f = f ∘ ϑ. In this paper, we investigate the relationship between properties of the symbol ϑ and the subnormality of the operators C ϑ and C ϑ ∗ . The main theorem shows, under a regularity hypothesis, that C ϑ ∗ is subnormal if and only if ϑ is in a restricted class of linear fractional transformations. The two previously known special cases, the affine transformations associated with the Cesaro operator and the automorphisms, are included as the extreme examples. The proof of the main theorem consists of verifying a moment condition and yields, in addition, the construction of a measure μ that exhibits C gj ∗ as multiplication by an analytic function on P 2( μ) and the identification of the minimal normal extension as a weighted sum of shifts.