Continuous casting apparatus comprising a plurality of compression units (11), each of which is defined by a lower roll (12) and an upper roll (13) configured to exert a compression action on a cast metal product (P). Each lower roll (12) defines with the respective upper roll (13) a passage gap (14) for the metal product (P). The passage gaps (14) of the compression units (11) are disposed aligned along a casting axis (Z) with an at least partly curved development. A straightening roll (15) is disposed on the extrados side of said casting axis (Z) and between at least two of the lower rolls (12).