In a circular knitting machine of the type having two needle beds or needle-carrying structures (1,8) for knitting plain and purl stitches, the lower needle-carrying structure (1) comprises a cylinder and the upper needle-carrying structure (8) has a substantially bell-like shape and needle tricks (11) of arcuate shape for correspondingly arcuately bent needles (13). The arcuately bent needles (13) are controlled by selecting jacks (12), in turn controlled by cams (18) arranged around the substantially bell-like structure (8). Such a machine can have the same number of needles (3,13) on both needle-carrying structures (1,8), and accordingly, a higher fineness for a given diameter than traditional cylinder and dial machines. It also enables knitting loops to be transferred between all of the needles (3,13) in the two structures (1,8), ' both from the purl stitch side to the plain and from the plain stitch side to the purl, thus enabling the knitting of various types of rib pattern, such as 1:1, 2:2, etc., without requiring any jack replacement.