  We present a survey article about the geometry of convex bodies on the
$d$-dimensional sphere $S^d$. We concentrate on the results based on the notion
of the width of a convex body $C \subset S^d$ determined by a supporting
hemisphere of $C$. Important tools are the lunes containing $C$. The supporting
hemispheres take over the role of the supporting half-spaces of a convex body
in Euclidean space, and lunes the role of strips. Also essential is the notion
of thickness of $C$, i.e, its minimum width. In particular, we describe
properties of reduced spherical convex bodies and spherical bodies of constant
width. The last notion coincides with the notions of complete bodies and bodies
of constant diameter on $S^d$. The reminded and commented here results concern
mostly the width, thickness, diameter, perimeter, area and extreme points of
spherical convex bodies, reduced bodies and bodies of constant width.
