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We introduce the MUSE gAlaxy Groups in COSMOS (MAGIC) survey, which was built to study the impact of environment on galaxy evolution over the last 8 Gyr. It consists of 17 MUSE fields targeting 14 massive structures at intermediate redshift ($0.3<z<0.8$) in the COSMOS area. We securely measured the redshifts for 1419 sources and identified 76 galaxy pairs and 67 groups of at least 3 members using a friends-of-friends algorithm. The environment of galaxies is quantified from group properties, as well as from global and local density estimators. The MAGIC survey has increased the number of objects with a secure spectroscopic redshift over its footprint by a factor of about 5. Most of the new redshifts have apparent magnitudes in the $z^{++}$ band $z_{app}^{++}>21.5$. The spectroscopic redshift completeness is high: in the redshift range of [OII] emitters ($0.25 \le z < 1.5$), where most of the groups are found, it globally reaches a maximum of 80% down to $z_{app}^{++}=25.9$, and locally decreases from $\sim 100$% to $\sim50$% in magnitude bins from $z_{app}^{++}=23-24$ to $z_{app}^{++}=25.5$. We find that the fraction of quiescent galaxies increases with local density and with the time spent in groups. A morphological dichotomy is also found between bulge-dominated quiescent and disk-dominated star-forming galaxies. As environment gets denser, the peak of the stellar mass distribution shifts towards $M_*>10^{10}~M_\odot$, and the fraction of galaxies with $M_*<10^9~M_\odot$ decreases significantly, even for star-forming galaxies. We also highlight peculiar features such as close groups, extended nebulae, and a gravitational arc. Our results suggest that galaxies are preprocessed in groups of increasing mass before entering rich groups and clusters. We publicly release two catalogs containing the properties of galaxies and groups, respectively.