A tyre assembling apparatus (2) made up of two halves (3a, 3b) axially movable close to each other, on which a carcass sleeve (7) of cylindrical conformation is fitted. Each half (3a, 3b) comprises radial sectors (5) each provided with a radially expandable holding member (14, 15) to engage the carcass sleeve (7) close to an annular anchoring structure (10). Supporting members (23) associated with the holding members (14, 15) define a cylindrical supporting surface for the carcass plies (9) extended in a cylindrical conformation, thereby making application of under-belt strips (26) easier. Turning-up devices (31) comprising circumferentially distributed levers (32) turn the side edges (7a, 7b) of the carcass sleeve (7) up around the annular anchoring structures (10), after the halves (3a, 3b) have been moved close to each other to give the carcass sleeve (7) a toroidal conformation and engage it to a belt package (29) provided with a tread band (30).