Kontsevich's work on the deformation of Poisson manifolds suggests a certain approach to deformation problems in general (already outlined in works of Goldman and Millson on complex manifolds.) Namely, a structure to deform should be viewed as a solution of Maurer-Cartan equation on a certain governing differential graded Lie (dgLie) algebra. In fact, to a dgLie algebra corresponds a Deligne groupoid, describing deformations. After a review of necessary tools I will present a similar approach to deformations of Lie quasibialgebras. I will describe the governing dgLie algebra and write a generalization of the Deligne groupoid for this case. There are some surprises: for example, the Lie bracket of the dgLie algebra is symmetric (instead of being antisymmetric) but it should not stop us! The problem of quantization of Lie quasibialgebras is however still open.