A $textit{portrait}$ $mathcal{P}$ on $mathbb{P}^N$ is a pair of finite point sets $Ysubseteq{X}subsetmathbb{P}^N$, a map $Yto X$, and an assignment of weights to the points in $Y$. We construct a parameter space $operatorname{End}_d^N[mathcal{P}]$ whose points correspond to degree $d$ endomorphisms $f:mathbb{P}^Ntomathbb{P}^N$ such that $f:Yto{X}$ is as specified by a portrait $mathcal{P}$, and prove the existence of the GIT quotient moduli space $mathcal{M}_d^N[mathcal{P}]:=operatorname{End}_d^N//operatorname{SL}_{N+1}$ under the $operatorname{SL}_{N+1}$-action $(f,Y,X)^phi=bigl(phi^{-1}circ{f}circphi,phi^{-1}(Y),phi^{-1}(X)bigr)$ relative to an appropriately chosen line bundle. We also investigate the geometry of $mathcal{M}_d^N[mathcal{P}]$ and give two arithmetic applications.
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