We introduce localization and sheaves to define projective schemes, and in particular the projective n-space. Afterwards, we define closed subschemes of projective space and show that they arise from quotients of graded rings by homogeneous ideals. We then define the Hilbert function and Hilbert polynomial to determine several invariants of closed subschemes of projective space: their degree, dimension, and arithmetic genus. Finally, we provide numerous examples with explicit computations, finding the invariants of hypersurfaces, curves, the twisted cubic and more.