A notion of unfolding, or multi-parameter deformation, of CR singularities of real submanifolds in complex manifolds is proposed, along with a definition of equivalence of unfoldings under the action of a group of analytic transformations. In the case of real surfaces in complex $2$-space, deformations of elliptic, hyperbolic, and parabolic points are analyzed by putting the parameter-dependent real analytic defining equations into normal forms up to some order. For some real analytic unfoldings in higher codimension, the method of rapid convergence is used to establish real algebraic normal forms. Table of Contents: Introduction; Topological considerations; Local defining equations and transformations; A complexification construction; Real surfaces in $\mathbb{C}^2$; Real $m$-submanifolds in $\mathbb{C}^n, m < n$; Rapid convergence proof of the main theorem; Some other directions; Bibliography. (MEMO/205/962).