  A general concept of perfect nonradiating modes in dielectric nanoparticles
of an arbitrary shape is put forward. It is proved that such modes exist in
axisymmetric dielectric nanoparticles and have unlimited radiative Q-factors.
With smart tuning of the excitation beams, perfect modes appear as deep minima
in the scattered radiation spectra (up to complete disappearance), but at the
same time, they have a giant amplification of the fields inside the particle.
Such modes have no analogs and can be useful for the realization of
nanosensors, low threshold nanolasers, and other strong nonlinear effects in
nanoparticles.
