  Let $M = \Gamma \backslash G$ be a nilmanifold endowed with an invariant
complex structure. We prove that Kuranishi deformations of abelian complex
structures are all invariant complex structures, generalizing a result of C.
Maclaughlin, H. Pedersen, Y.S. Poon and S. Salamon for 2-step nilmanifolds. We
characterize small deformations that remain abelian. As an application, we
observe that at real dimension six, the deformation process of abelian complex
structures is stable within the class of nilpotent complex structures. We give
an example to show that this property does not hold in higher dimension.
