In this paper we consider covering problems in spherical geometry. Let cov_qSⁿ₁ be the smallest radius of q equal metric balls that cover n-dimensional unit sphere Sⁿ₁. We show that cov_qSⁿ₁ = π/2 for 2 ≤ q ≤ n + 1 and π - arccos(-1/(n+1)) for q = n + 2. The configuration of centers of balls realizing cov_qSⁿ₁ are established, simultaneously. Moreover, some properties of cov_qX for the compact metric apace X in general, are proved.