The minimal spherical cap dispersion disp_𝒞(n,d) is the largest number ε∈ (0,1] such that, no matter how n points are distributed on the d-dimensional Euclidean unit sphere 𝕊^d, there is always a spherical cap with normalized area ε not containing any of the points. We study the behavior of disp_𝒞(n,d) as n and d grow to infinity. We develop connections to the problems of sphere covering and approximation of the Euclidean unit ball by inscribed polytopes. Existing and new results are presented in a unified way. Upper bounds on disp_𝒞(n,d) result from choosing the points independently and uniformly at random and possibly adding some well-separated points to close large gaps. Moreover, we study dispersion with respect to intersections of caps.