We show that nonlinear tight-binding lattices of different geometries and dimensionalities, display an universal selftrapping behavior. First, we consider the single nonlinear impurity problem in various tight-binding lattices, and use the Green's function formalism for an exact calculation of the minimum nonlinearity strength to form a stationary bound state. For all lattices, we find that this critical nonlinearity parameter (scaled by the energy of the bound state), in terms of the nonlinearity exponent, falls inside a narrow band, which converges to e^(1/2) at large exponent values. Then, we use the Discrete Nonlinear Schroedinger (DNLS) equation to examine the selftrapping dynamics of a single excitation, initially localized on the single nonlinear site, and compute the critical nonlinearity parameter for abrupt dynamical selftrapping. For a given nonlinearity exponent, this critical nonlinearity, properly scaled, is found to be nearly the same for all lattices. Same results are obtained when generalizing to completely nonlinear lattices, suggesting an underlying selftrapping universality behavior for all nonlinear (even disordered) tight-binding lattices described by DNLS.