We define functions of noncommuting self-adjoint operators with the help of double operator integrals. We are studying the problem to find conditions on a function f on R², for which the map (A, B)↦f(A, B) is Lipschitz in the operator norm and in Schatten-von Neumann norms S_p. It turns out that for functions f in the Besov class B_∞,1¹(R²), the above map is Lipschitz in the S_p norm for p∈[1, 2]. However, it is not Lipschitz in the operator norm, nor in the S_p norm for p>2. The main tool is triple operator integrals. To obtain the results, we introduce new Haagerup-like tensor products of L^∞ spaces and obtain Schatten-von Neumann norm estimates of triple operator integrals. We also obtain similar results for functions of noncommuting unitary operators.