Summability of spherical h-harmonic expansions with respect to the weight function Q d jΒΌ1 jx j j 2kj Γ°k j X0Γ on the unit sphere S dΓ1 is studied. The main result characterizes the critical index of summability of the Cesaro Γ°C; dΓ means of the h-harmonic expansion; it is proved that the Γ°C; dΓ means of any continuous function converge uniformly in the norm of CΓ°S dΓ1 Γ if and only if d4Γ°d Γ 2Γ=2 ΓΎ P d jΒΌ1 k j Γ min 1pjpd k j : Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and S dΓ1 ; the Γ°C; dΓ means of the h-harmonic expansion of a continuous function f converges pointwisely to f if d4Γ°d Γ 2Γ=2: Similar results are established for the orthogonal expansions with respect to the weight functions Q d jΒΌ1 jx j j 2kj Γ°1 Γ jxj 2 Γ mΓ1=2 on the unit ball B d and Q d jΒΌ1 x kj Γ1=2 j Γ°1 Γ jxj 1 Γ mΓ1=2 on the simplex T d : As a related result, the Cesaro summability of the generalized Gegenbauer expansions associated to the weight function jtj 2m Γ°1 Γ t 2 Γ lΓ1=2 on Β½Γ1; 1 is studied, which is of interest in itself.