  It is commonly held that a necessary condition for the existence of solitons
in nonlinear-wave systems is that the soliton's frequency (spatial or temporal)
must not fall into the continuous spectrum of radiation modes. However, this is
not always true. We present a new class of {\it % codimension-one} solitons
(i.e., those existing at isolated frequency values) that are {\em embedded}
into the continuous spectrum. This is possible if the spectrum of the
linearized system has (at least) two branches, one corresponding to
exponentially localized solutions, and the other to radiation modes. An
embedded soliton (ES) is obtained when the latter component exactly vanishes in
the solitary-wave's tail. The paper contains both a survey of recent results
obtained by the authors and some new results, the aim being to draw together
several different mechanism underlying the existence of ESs. We also consider
the distinctive property of {\em semi-stability} of ES, and {\em moving} ESs.
Results are presented for four different physical models, including an extended
5th-order KdV equation describing surface waves in inviscid fluids, and three
models from nonlinear optics. One of them pertains to a resonant Bragg grating
in an optical fiber with a cubic nonlinearity, while two others describe
second-harmonic generation (SHG) in the temporal or spatial domain (i.e.,
respectively, propagating pulses in nonlinear optical fibers, or stationary
patterns in nonlinear planar waveguides). Special attention is paid to the SHG
model in the temporal domain for a case of competing quadratic and cubic
nonlinearities. An essential new result is that ES is, virtually, fully stable
in the latter model in the case when both harmonics have anomalous dispersion.
