  We analyze the Gottlieb groups of function spaces. Our results lead to
explicit decompositions of the Gottlieb groups of many function spaces
map(X,Y)---including the (iterated) free loop space of Y---directly in terms of
the Gottlieb groups of Y. More generally, we give explicit decompositions of
the generalized Gottlieb groups of map(X,Y) directly in terms of generalized
Gottlieb groups of Y. Particular cases of our results relate to the torus
homotopy groups of Fox. We draw some consequences for the classification of
T-spaces and G-spaces. For X, Y finite and Y simply connected, we give a
formula for the ranks of the Gottlieb groups of map(X,Y) in terms of the Betti
numbers of X and the ranks of the Gottlieb groups of Y. Under these hypotheses,
the Gottlieb groups of map(X,Y) are finite groups in all but finitely many
degrees.
