Lovász and Yemini proved in 1982 that every 6-connected graph is rigid in the plane, and they conjectured that for any dimension d, every d(d+1)-connected graph is rigid in d-dimensional space. In this talk I will describe a probabilistic method that proves this conjecture. The same approach can be used to show that d(d+1)-connected graphs are also globally rigid, confirming a conjecture of Connelly, Jordán and Whiteley. This stronger result is obtained using the notion of weakly globally linked vertex pairs.