Let X be a Gorenstein Fano threefold with at most canonical singularities.  
It is known that there are only finitely many deformation families of such
X, so one may ask for a complete classification as was done in the smooth
case by Iskovskikh, Mori and Mukai. An important question is under which
conditions X arises as a degeneration of a smooth Fano threefold, and if
that is the case, how X and its "smoothing" are related. In 1997 Namikawa
proved the existence of a smoothing, if X has only terminal singularities,
i.e., X is the special fiber of a flat family Z -> D with general fiber
Z_t a smooth Fano threefold. Here Z is an irreducible complex space, not
necessarily smooth. We show that the Picard groups of X and Z_t are
isomorphic in the terminal case and give some examples concerning
canonical singularities. Here a smoothing need not exist, and even if it
exists, the Picard number may jump.