Maps between non-commutative spaces

We examine maps between noncommutative projective spaces. A surjection of graded rings A-->A/J induces a closed immersion Proj(A/J)-->Proj(A). A homomorphism f:A-->B between graded rings induces an affine map U --> Proj(A) from a non-empty open subspace U of Proj(B). If A^{(n)} is the n-th Veronese subalgebra of a graded ring A there is a map Proj(A)-->Proj(A^{(n)}); we identify open subspaces on which this map is an isomorphism. Applying these results when A is (a quotient of) a weighted polynomial ring produces a non-commutative resolution of (a closed subscheme of) a weighted projective space.