We construct the moduli spaces of stable maps, Mg,n(P r , d), via geometric invariant theory (GIT). This construction is only valid over Spec C, but a special case is a GIT presentation of the moduli space of stable curves of genus g with n marked points, Mg,n; this is valid over Spec Z. In another paper by the first author, a small part of the argument is replaced, making the result valid in far greater generality. Our method follows that used in the case n = 0 by Gieseker in [7], to construct Mg , though our proof that the semistable set is nonempty is entirely different.Proof. IfJ ss (L) = J, thenJ// L SL(W ) is a categorical quotient of J, and if J ss (L) =J s (L), the quotient is an orbit space. The result follows from Proposition 3.4 and Proposition 2.3.In the following GIT construction, we will first seek a linearisation L such that J ss (L) ⊂ J. This has many useful implications, which we shall explore now. In particular, it gives us the first half of the desired equality:J ss (L) =J s (L).Proposition 3.6. Suppose there exists a linearisation L such thatJ ss (L) ⊆ J. ThenJ ss (L) =J s (L).Proof. Every point of J corresponds to a moduli stable map, and so has finite stabiliser. The result follows from Corollary 2.6. This leaves us with the second required equality, thatJ ss = J. In this paper, we shall prove it using the existing construction of M g,n (P r , d) over C, given by Fulton and Pandharipande in [6]. This is not necessary (see [2]), but for brevity we take this shortcut for now.Corollary 3.7. There exists a map j : J → M g,n (P r , d), which is an orbit space for the SL(W ) action, and in particular a categorical quotient. The morphism j is universally closed.