The preeminent merit of linear or nonlinear recursive filters is their ability to represent unknown systems with far fewer coefficients than their nonrecursive counterparts. Unfortunately, nonlinear recursive filters become, in general, unstable for large amplitudes of the input signal. In this paper, we introduce a novel subclass of the linear-in-the-parameters nonlinear recursive filters whose members are stable according to the bounded-input bounded-output criterion. It is shown that these filters, called stable recursive functional link polynomial filters, are universal approximators for causal, time-invariant, infinite-memory, continuous, nonlinear systems according to the Stone-Weierstrass theorem. To further reduce the filter complexity, simplified structures are also investigated. Even though simplified filters are no more universal approximators, good performance is demonstrated exploiting either data measured on real systems or available in the literature as benchmarks for nonlinear system identification.