We define Drinfeld orbifold algebras as filtered algebras deforming the skew group algebra (semidirect product) arising from the action of a finite group on a polynomial ring. They simultaneously generalizeWeyl algebras, graded (or Drinfeld) Hecke algebras, rational Cherednik algebras, symplectic reflection algebras, and universal enveloping algebras of Lie algebras with group actions. Wegive necessary and sufficient conditions on defining parameters to obtain Drinfeld orbifold algebras in two general formats, both algebraic and homological. Our algebraic conditions hold over any field of characteristic other than two, including fields whose characteristic divides the order of the acting group. We explain the connection between Hochschild cohomology and a Poincar-Birkhoff-Witt property explicitly (using Gerstenhaber brackets). We also classify those deformations of skew group algebras which arise as Drinfeld orbifold algebras and give applications for abelian groups.