  The NASA $Kepler$ mission obtained long-term high-quality photometric
observations for a large number of stars in its original field of view from
2009 to 2013. In order to provide reliable stellar parameters in a homogeneous
way, the LAMOST telescope began to carry out low-resolution spectroscopic
observations for as many stars as possible in the $Kepler$ field in 2012. By
September 2018, 238,386 low-resolution spectra with SNR$_g \geq 6$ had been
collected for 155,623 stars in the $Kepler$ field, enabling the determination
of atmospheric parameters and radial velocities, as well as spectral
classification of the target stars. This information has been used by
astronomers to carry out research in various fields, including stellar
pulsations and asteroseismology, exoplanets, stellar magnetic activity and
flares, peculiar stars and the Milky Way, binary stars, etc. We summarize the
research progress in these fields where the usage of data from the
LAMOST-$Kepler$ (LK) project has played a role. In addition, time-domain
medium-resolution spectroscopic observations have been carried out for about
12,000 stars in four central plates of the $Kepler$ field since 2018. The
currently available results show that the LAMOST-$Kepler$ medium resolution
(LK-MRS) observations provide qualified data suitable for research in
additional science projects including binaries, high-amplitude pulsating stars,
etc. As LAMOST is continuing to collect both low- and medium-resolution spectra
of stars in the $Kepler$ field, we expect more data to be released continuously
and new scientific results to appear based on the LK project data.
