  The intersection $L$ of two different non-opposite hemispheres $G$ and $H$ of
a $d$-dimensional sphere $S^d$ is called a lune. By the thickness of $L$ we
mean the distance of the centers of the $(d-1)$-dimensional hemispheres
bounding $L$. For a hemisphere $G$ supporting a %spherical convex body $C
\subset S^d$ we define ${\rm width}_G(C)$ as the thickness of the narrowest
lune or lunes of the form $G \cap H$ containing $C$. If ${\rm width}_G(C) =w$
for every hemisphere $G$ supporting $C$, we say that $C$ is a body of constant
width $w$. We present properties of these bodies. In particular, we prove that
the diameter of any spherical body $C$ of constant width $w$ on $S^d$ is $w$,
and that if $w < \frac{\pi}{2}$, then $C$ is strictly convex. Moreover, we are
checking when spherical bodies of constant width and constant diameter
coincide.
