Noncommutative Gröbner bases and Hochschild cohomology. We present a computational approach for finding Hochschild cohomology groups of a finite-dimensional algebra Λ. Our approach utilizes a lifted Gröbner basis to the enveloping algebra Λ e of Λ. This construction has been implemented in a program ENVELOPE which can be used in conjunction with an existing resolution package GRB. Together, these programs allow one to compute minimal projective resolutions of Λ over Λ e . The resolutions can then be used to construct cohomology groups. We report on some of the computations that have already been completed using the techniques from this paper.