We construct and study noncommutative deformations of toric varieties by
combining techniques from toric geometry, isospectral deformations, and
noncommutative geometry in braided monoidal categories. Our approach utilizes
the same fan structure of the variety but deforms the underlying embedded
algebraic torus. We develop a sheaf theory using techniques from noncommutative
algebraic geometry. The cases of projective varieties are studied in detail,
and several explicit examples are worked out, including new noncommutative
deformations of Grassmann and flag varieties. Our constructions set up the
basic ingredients for thorough study of instantons on noncommutative toric
varieties, which will be the subject of the sequel to this paper.