  Unconventional superconductivity in molecular conductors is observed at the
border of metal-insulator transitions in correlated electrons under the
influence of geometrical frustration. The symmetry as well as the mechanism of
the superconductivity (SC) is highly controversial. To address this issue, we
theoretically explore the electronic properties of carrier-doped molecular Mott
system $\kappa$-(BEDT-TTF)$_2$X. We find significant electron-hole doping
asymmetry in the phase diagram where antiferromagnetic (AF) spin order,
different patterns of charge order, and SC compete with each other. Hole-doping
stabilizes AF phase and promotes SC with "$d_{xy}$"-wave symmetry, which has
similarities with high-$T_c$ cuprates. In contrast, in the electron-doped side,
geometrical frustration destabilizes the AF phase and the enhanced charge
correlation induces another SC with "extended-$s$+$d_{x^2-y^2}$"-wave symmetry.
Our results disclose the mechanism of each phase appearing in filling-control
molecular Mott systems, and elucidate how physics of different
strongly-correlated electrons are connected, namely, molecular conductors and
high-$T_c$ cuprates.
