  The nuclear fusion is a reaction to form a compound nucleus. It plays an
important role in several circumstances in nuclear physics as well as in
nuclear astrophysics, such as synthesis of superheavy elements and
nucleosynthesis in stars. Here we discuss two recent theoretical developments
in heavy-ion fusion reactions at energies around the Coulomb barrier. The first
topic is a generalization of the Wong formula for fusion cross sections in a
single-channel problem. By introducing an energy dependence to the barrier
parameters, we show that the generalized formula leads to results practically
indistinguishable from a full quantal calculation, even for light symmetric
systems such as $^{12}$C+$^{12}$C, for which fusion cross sections show an
oscillatory behavior. We then discuss a semi-microscopic modeling of heavy-ion
fusion reactions, which combine the coupled-channels approach to the
state-of-the-art nuclear structure calculations for low-lying collective
motions. We apply this method to subbarrier fusion reactions of
$^{58}$Ni+$^{58}$Ni and $^{40}$Ca+$^{58}$Ni systems, and discuss the role of
anharmonicity of the low-lying vibrational motions.
