  We introduce a new construction of a surface link in the 4-space. We
construct a surface link as a branched covering over the standard torus, which
we call a torus-covering link. We show that a certain torus-covering $T^2$-link
is equivalent to the split union of spun $T^2$-links and turned spun
$T^2$-links. We show that a certain torus-covering $T^2$-link has a
non-classical link group. We give a certain class of ribbon torus-covering
$T^2$-links. We present the quandle cocycle invariant of a certain
torus-covering $T^2$-link obtained from a classical braid, by using the quandle
cocycle invariants of the closure of the braid.
