An apparatus for packaging a product arranged on a support, said support having a base wall and side wall, said apparatus comprising: a frame (10; 601); a transport assembly (2; 604) arranged on the frame (10; 610) and configured for displacing one or more supports along a predefined path; a packaging assembly (5; 606) carried by said frame (10; 601) and operative at a packaging station (201; 605) positioned along said path, the packaging assembly (5; 606) being configured for tightly fixing a film sheet to said one or more supports, wherein the packaging assembly (5; 606) comprises a base unit, configured for receiving and holding the at least one support, and a top unit, operating above the base unit and configured for positioning the film sheet above the base unit; at least one perforating unit carried by the frame and operating in correspondence of the packaging station, the perforating unit being configured to form at least one hole in the side wall of the at least one support present in the packaging station, the perforating unit having a perforating tool operable between a rest position, where the tool is spaced apart from the side wall of said support, and an operating position, where the tool acts on said support and forms at least one though hole in said side wall, wherein the perforating tool has a pointed tip configured for progressively contacting the side wall of the support upon displacement of the perforating tool from the rest position to the operating position; a vacuum arrangement configured for removing air from within said at least one support underneath said film sheet at least in part through said at least one hole; wherein the perforating tool of each perforating unit defines a tip aperture connected to a suction aperture (110; 105b) spaced from the tip aperture and in fluid communication with said vacuum arrangement in order to remove air from within said at least one support when the perforating tool pointed tip is inserted in said at least one hole; and further wherein the base unit (53; 53') defines a chamber configured for housing said at least one support and presents passages (530a, 530b; 530a', 530b'), either on its side wall or on its bottom wall, designed to place into fluid communication the vacuum arrangement and a volume (300; 300') defined at least between an external surface of the at least one support and an inner surface of the base unit (53; 53'). a frame (10; 601); a transport assembly (2; 604) arranged on the frame (10; 610) and configured for displacing one or more supports along a predefined path; a packaging assembly (5; 606) carried by said frame (10; 601) and operative at a packaging station (201; 605) positioned along said path, the packaging assembly (5; 606) being configured for tightly fixing a film sheet to said one or more supports, wherein the packaging assembly (5; 606) comprises a base unit, configured for receiving and holding the at least one support, and a top unit, operating above the base unit and configured for positioning the film sheet above the base unit; at least one perforating unit carried by the frame and operating in correspondence of the packaging station, the perforating unit being configured to form at least one hole in the side wall of the at least one support present in the packaging station, the perforating unit having a perforating tool operable between a rest position, where the tool is spaced apart from the side wall of said support, and an operating position, where the tool acts on said support and forms at least one though hole in said side wall, wherein the perforating tool has a pointed tip configured for progressively contacting the side wall of the support upon displacement of the perforating tool from the rest position to the operating position; a vacuum arrangement configured for removing air from within said at least one support underneath said film sheet at least in part through said at least one hole;