In the talk I will review the relation between minimally supersymmetric gauge theories in four and three dimensions, and singulatiries in G2 and Spin(7) manifolds, respectively. These minimally supersymmetric gauge theories exhibit very rich dynamical structure, including such phenomesna as dynamical (super)symmetry breaking, parity symmetry breaking, various phases, and transitions between them. All these phenomena find their counterparts in the geometry of exceptional holonomy manifolds, which is also extremely rich. A careful study of various models will draw our attention to Spin(7) manifolds, which turn out to be a natural arena for anomalies in string theory and in M-theory.