An apparatus for packaging a product arranged on a support, said support optionally having a base wall and a side wall, said apparatus comprising: a film supplying assembly configured for supplying at least one film; a packaging assembly configured for tightly closing said one or more supports with said film, the packaging assembly including: - at least one lower tool comprising a prefixed number of seats for receiving said one or more supports, and - at least one upper tool (21; 21', 21 ") cooperating with the lower tool and configured for holding at least a portion (10b; 18a) of said film; wherein the upper tool (21; 21', 21") and the lower tool are configured to be movable the one relative to the other between at least a first operating condition, where the lower tool and the upper tool allow positioning of the one or more supports at said seats, and a second operating condition, where the lower tool and the upper tool delimit a packaging chamber; wherein the upper tool (21; 21', 21") comprises: o a head having a respective active surface facing the one or more supports and configured for receiving the film portion (10b; 18a) of said film, o holding means associated to the head and configured for attracting the film portion (10b; 18a) of film towards said active surface, o a peripheral body positioned around the head, the head and the peripheral body being mounted for relative motion the one with respect to the other among a plurality of relative positions and being configured for defining a cavity delimited by said active surface and by an inner wall of said peripheral body; and a control unit connected to the packaging assembly and configured for executing a film deformation procedure comprising the steps of: - commanding the relative motion of the peripheral body with respect to the head in order to define said cavity, - activating the holding means, - wherein said steps of commanding the relative motion and activating the holding means are coordinated by the control unit such as to cause said film portion (10b; 18a) to move from a substantially flat configuration to a substantially tri-dimensional configuration inside said cavity. a film supplying assembly configured for supplying at least one film; a packaging assembly configured for tightly closing said one or more supports with said film, the packaging assembly including: - at least one lower tool comprising a prefixed number of seats for receiving said one or more supports, and - at least one upper tool (21; 21', 21 ") cooperating with the lower tool and configured for holding at least a portion (10b; 18a) of said film; wherein the upper tool (21; 21', 21") and the lower tool are configured to be movable the one relative to the other between at least a first operating condition, where the lower tool and the upper tool allow positioning of the one or more supports at said seats, and a second operating condition, where the lower tool and the upper tool delimit a packaging chamber; wherein the upper tool (21; 21', 21") comprises: o a head having a respective active surface facing the one or more supports and configured for receiving the film portion (10b; 18a) of said film, o holding means associated to the head and configured for attracting the film portion (10b; 18a) of film towards said active surface, o a peripheral body positioned around the head, the head and the peripheral body being mounted for relative motion the one with respect to the other among a plurality of relative positions and being configured for defining a cavity delimited by said active surface and by an inner wall of said peripheral body; and - at least one lower tool comprising a prefixed number of seats for receiving said one or more supports, and - at least one upper tool (21; 21', 21 ") cooperating with the lower tool and configured for holding at least a portion (10b; 18a) of said film; o a head having a respective active surface facing the one or more supports and configured for receiving the film portion (10b; 18a) of said film, o holding means associated to the head and configured for attracting the film portion (10b; 18a) of film towards said active surface, o a peripheral body positioned around the head, the head and the peripheral body being mounted for relative motion the one with respect to the other among a plurality of relative positions and being configured for defining a cavity delimited by said active surface and by an inner wall of said peripheral body; and a control unit connected to the packaging assembly and configured for executing a film deformation procedure comprising the steps of: - commanding the relative motion of the peripheral body with respect to the head in order to define said cavity, - activating the holding means, - wherein said steps of commanding the relative motion and activating the holding means are coordinated by the control unit such as to cause said film portion (10b; 18a) to move from a substantially flat configuration to a substantially tri-dimensional configuration inside said cavity. - commanding the relative motion of the peripheral body with respect to the head in order to define said cavity, - activating the holding means, - wherein said steps of commanding the relative motion and activating the holding means are coordinated by the control unit such as to cause said film portion (10b; 18a) to move from a substantially flat configuration to a substantially tri-dimensional configuration inside said cavity.