Abstract: Deformation theory is a part of algebraic geometry which studies how algebraic varieties deform. In this talk I will report on joint work in progress with Nathan Ilten (Simon Fraser University) and Francesco Meazzini (Sapienza Università di Roma) about deformations of affine schemes which are defined by monomial ideals. I will start by recalling the basics of deformation theory and I will try to explain the rich combinatorial structure of monomial ideals to produce resolutions and compute cotangent cohomology.