  We establish trace theorems for function spaces defined on general Ahlfors
regular metric spaces $Z$. The results cover the Triebel-Lizorkin spaces and
the Besov spaces for smoothness indices $s<1,$ as well as the first order
Haj{\l}asz-Sobolev space $M^{1,p}(Z)$. They generalize the classical results
from the Euclidean setting, since the traces of these function spaces onto any
closed Ahlfors regular subset $F \subset Z$ are Besov spaces defined
intrinsically on $F$. Our method employs the definitions of the function spaces
via hyperbolic fillings of the underlying metric space.
