Resumo: The limits of Weierstrass points on stable curves of genus 3 (even of non-compact type) are studied. More precisely, one tries to solve the following general problem: determine all the zero-dimensional subschemes of a stable singular curve that are limits of Weierstrass points on "nearby" smooth curves. For stable curves of genus three, the possible limits may be guessed by looking at suitable models of irreducible plane quartics, relating to the theory of Widland-Lax of Weierstrass points on Gorenstein irreducible curves. Some open problems will be discussed.