  We report the discovery of exotic superconductivity (SC) and novel magnetism
in heavy-fermion (HF) compounds, CeCu$_2$Si$_2$, CeRhIn$_5$ and CeIn$_3$
through nuclear-quadrupole-resonance (NQR) measurements under pressure ($P$).
The exotic SC in a homogeneous CeCu$_2$Si$_2$ revealed antiferromagnetic
critical fluctuations at the border to antiferromagnetism (AFM) or marginal
AFM. The uniform mixed phase of SC and AFM in CeCu$_2$(Si$_{1-x}$Ge$_x$)$_2$
emerges on a microscopic level, once a tiny amount of 1%Ge($x=0.01$) is
substituted for Si to expand its lattice. The application of minute pressure
($P\sim 0.19$ GPa) suppresses the sudden emergence of the AFM caused by doping
Ge. The persistence of the low-lying magnetic excitations at temperatures lower
than $T_c$ and $T_N$ is ascribed due to the uniform mixed phase of SC and AFM.
Likewise, the $P$-induced HF superconductor CeRhIn$_5$ coexists with AFM on a
microscopic level in $P = 1.5$ - 1.9 GPa. The unconventional gapless nature of
SC in the low-lying excitation spectrum emerges due to the uniform mixed phase
of AFM and SC. By contrast, in CeIn$_3$, we propose that the magnetic
excitations such as spin-density fluctuations induced by the first-order phase
transition from the AFM to the paramagnetism (PM) might mediate attractive
interaction to form the Cooper pairs in the novel phase of AFM.
