A family of spherical caps of the 2-dimensional unit sphere is called a totally separable packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a totally separable packing in the 2-dimensional unit sphere. The talk surveys the status of this problem and its dual on thinnest totally separable coverings. Discrete geometry methods are combined with convexity ones. This is a joint work with Z. Langi (Tech. Univ., Budapest).