  Harmonic lasing in a free electron laser with a planar undulator (under the
condition that the fundamental frequency is suppressed) might be a cheap and
efficient way of extension of wavelength ranges of existing and planned X-ray
FEL facilities. Contrary to nonlinear harmonic generation, harmonic lasing can
provide much more intense, stable, and narrow-band FEL beam which is easier to
handle due to the suppressed fundamental frequency. In this paper we perform a
parametrization of the solution of the eigenvalue equation for lasing at odd
harmonics, and present an explicit expression for FEL gain length, taking into
account all essential effects. We propose and discuss methods for suppression
of the fundamental harmonic. We also suggest a combined use of harmonic lasing
and lasing at the retuned fundamental wavelength in order to reduce bandwidth
and to increase brilliance of X-ray beam at saturation. Considering 3rd
harmonic lasing as a practical example, we come to the conclusion that it is
much more robust than usually thought, and can be widely used in the existing
or planned X-ray FEL facilities. In particular, LCLS after a minor modification
can lase to saturation at the 3rd harmonic up to the photon energy of 25-30 keV
providing multi-gigawatt power level and narrow bandwidth. As for the European
XFEL, harmonic lasing would allow to extend operating range (ultimately up to
100 keV), to reduce FEL bandwidth and to increase brilliance, to enable
two-color operation for pump-probe experiments, and to provide more flexible
operation at different electron energies. Finally, in this paper we discover
that in a part of the parameter space, corresponding to the operating range of
soft X-ray beamlines of X-ray FEL facilities (like SASE3 beamline of the
European XFEL), harmonics can grow faster than the fundamental wavelength.
