  We present the KMOS Galaxy Evolution Survey (KGES), a $K$-band Multi-Object
Spectrograph (KMOS) study of the H$\alpha$ and [NII] emission from 288 $K$
band-selected galaxies at $1.2 \lesssim z \lesssim 1.8$, with stellar masses in
the range $\log_{10}(M_{*}/\rm{M}_{\odot})\approx$9-11.5. In this paper, we
describe the survey design, present the sample, and discuss the key properties
of the KGES galaxies. We combine KGES with appropriately matched samples at
lower redshifts from the KMOS Redshift One Spectroscopic Survey (KROSS) and the
SAMI Galaxy Survey. Accounting for the effects of sample selection, data
quality, and analysis techniques between surveys, we examine the kinematic
characteristics and angular momentum content of star-forming galaxies at
$z\approx1.5$, $\approx1$ and $\approx0$. We find that stellar mass, rather
than redshift, most strongly correlates with the disc fraction amongst
star-forming galaxies at $z \lesssim 1.5$, observing only a modest increase in
the prevalence of discs between $z\approx1.5$ and $z\approx0.04$ at fixed
stellar mass. Furthermore, typical star-forming galaxies follow the same median
relation between specific angular momentum and stellar mass, regardless of
their redshift, with the normalisation of the relation depending more strongly
on how disc-like a galaxy's kinematics are. This suggests that massive
star-forming discs form in a very similar manner across the $\approx$ 10 Gyr
encompassed by our study and that the inferred link between the angular
momentum of galaxies and their haloes does not change significantly across the
stellar mass and redshift ranges probed in this work.
