For d>= 1, it is an easy fact that a graph G must have minimum degree at least d in order to be rigid in R^d. In the talk I will present a recent work, joint with A. Lew, E. Nevo and Y. Peled, where we show that the threshold probability for a random graph G(n, p) to be d-rigid, coincides with the known threshold for having minimum degree d. This extends the classical proof for a random graph to be connected, which corresponds to the case d=1.