It is described an apparatus (1) for packaging a product (P) comprising: a forming station (2) configured for receiving a base film (3) and for forming with it a precursor body (4) having a plurality of tray-shaped elements (5) adjacent to each other, and a packaging station (8) configured for receiving said precursor body (4) and a closing film (9), the packaging station (8) being configured for attaching the closing film (9) to the precursor body (4) at least at an upper opening (5a) of a predetermined number of said tray-shaped elements (5) for forming packages (C) flanked to each other. A movement device (13) is configured for moving at least one of said forming (2) and packaging stations (8), for conferring to the precursor body (4) a step-by-step discontinuous movement along a predetermined advancement path (T). It is also described a packaging process using the apparatus (1).