After brief explanation of the basic definitions and notions connected with nonlocal symmetries of DEs the authors present a systematic method to derive the nonlocal symmetries for PDEs with two independent variables and show that KdV \(\frac{\partial u}{\partial t}=u\frac{\partial u}{\partial x}+\frac{\partial ^3 u}{\partial x^3}\) and Burgers \(\frac{\partial u}{\partial t}=\frac{\partial ^2 u}{\partial x^2}+u\frac{\partial u}{\partial x}\) equations admit a sequence of nonlocal symmetries. Then the method is extended to construct the nonlocal symmetries for partial differential-difference equations (PDDEs) and it is shown that the Volterra \(\frac{\partial u_n}{\partial t}=u_n(u_{n+1}-u_{n-1})\) and relativistic Toda lattice \(\frac{\partial u_n}{\partial t}=u_n(u_{n-1}-u_{n+1}+v_{n}-v_{n+1}), \frac{\partial v_n}{\partial t}=v_n(u_{n-1}-u_{n})\) equations also admit a sequence of nonlocal symmetries. The obtained nonlocal symmetries are used to derive the recursion (matrix recursion) operators for the indicated equations. For the last two equations a sequence of nonlocal conserved densities is found. However it is observed that the existence of nonlocal symmetries and relevant recursion operators of DEs and PDDEs does not always determine their mathematical structures, for example, the bi-Hamiltonian representation. The problem is posed to extend this method to nonlinear evolution equations with three independent variables.