Let X be compact Hausdorff, and ϕ : X → X a continuous surjection. Let A be the semicrossed product algebra corresponding to the relation f U = U f • ϕ or to the relation U f = f • ϕU. Then the C * -envelope of A is the crossed product of a commutative C * -algebra which contains C(X) as a subalgebra, with respect to a homeomorphism which we construct. We also show there are"sufficiently many" nest representations.