Let H denote the class of all normalized complex-valued harmonic functions f = h + g in the unit disk D, and let S 0 H denote the class of univalent and sense-preserving functions f in H such that f z (0) = 0. If K = H + G denotes the harmonic Koebe function whose dilation isHere, a n , b n , A n , and B n denote the Maclaurin coefficients of h, g, H, and G. We show that the radius of univalence of the family F is 0.112903 . . .. We also show that this number is also the radius of the fully starlikeness of F . Analogous results are proved for a family which contains the class of harmonic convex functions in H. We use the new coefficient estimate for bounded harmonic mappings and Lemma 1.6 to improve Bloch-Landau constant for bounded harmonic mappings.