In this paper we study the $5$ families of genus $3$ compact Riemann surfaces which are normal coverings of the Riemann sphere branched over $4$ points from very different aspects: their moduli spaces, the uniform Belyi functions that factorize through the quotient by the automorphism groups and the Weierstrass points of the non hyperelliptic families.
"Genus 3 normal coverings of the Riemann sphere branched over 4 points." Rev. Mat. Iberoamericana 22 (2) 413 - 454, September, 2006.