We present the KMOS-CLASH (K-CLASH) survey, a K-band Multi-Object Spectrograph (KMOS) survey of the spatially-resolved gas properties and kinematics of 191 (predominantly blue) H$alpha$-detected galaxies at $0.2 lesssim z lesssim 0.6$ in field and cluster environments. K-CLASH targets galaxies in four Cluster Lensing And Supernova survey with Hubble (CLASH) fields in the KMOS $IZ$-band, over $7$ radius ($approx2$-$3$ Mpc) fields-of-view. K-CLASH aims to study the transition of star-forming galaxies from turbulent, highly star-forming disc-like and peculiar systems at $zapprox1$-$3$, to the comparatively quiescent, ordered late-type galaxies at $zapprox0$, and to examine the role of clusters in the build-up of the red sequence since $zapprox1$. In this paper, we describe the K-CLASH survey, present the sample, and provide an overview of the K-CLASH galaxy properties. We demonstrate that our sample comprises star-forming galaxies typical of their stellar masses and epochs, residing both in field and cluster environments. We conclude K-CLASH provides an ideal sample to bridge the gap between existing large integral-field spectroscopy surveys at higher and lower redshifts. We find that star-forming K-CLASH cluster galaxies at intermediate redshifts have systematically lower stellar masses than their star-forming counterparts in the field, hinting at possible downsizing scenarios of galaxy growth in clusters at these epochs. We measure no difference between the star-formation rates of H$alpha$-detected, star-forming galaxies in either environment after accounting for stellar mass, suggesting that cluster quenching occurs very rapidly during the epochs probed by K-CLASH, or that star-forming K-CLASH galaxies in clusters have only recently arrived there, with insufficient time elapsed for quenching to have occured.
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