  In this proceeding, recent theoretical investigations by the authors on the
multiferroic $R$MnO$_3$ perovskites are briefly reviewed at first. Using the
double-exchange model, the realistic spiral spin order in undoped manganites
such as TbMnO$_3$ and DyMnO$_3$ is well reproduced by incorporating a weak
next-nearest neighbor superexchange ($\sim10%$ of nearest neighbor
superexchange) and moderate Jahn-Teller distortion. The phase transitions from
the A-type antiferromagnet (as in LaMnO$_3$), to the spiral phase (as in
TbMnO$_3$), and finally to the E-type antiferromagnet (as in HoMnO$_3$), with
decreasing size of the $R$ ions, were also explained. Moreover, new results of
phase diagram of the three-dimensional lattice are also included. The
ferromagnetic tendency recently discovered in the LaMnO$_3$ and TbMnO$_3$ thin
films is explained by considering the substrate stress. Finally, the
relationship between our double-exchange model and a previously used
$J_1$-$J_2$-$J_3$ model is further discussed from the perspective of spin wave
excitations.
