Based on the definitions of quantum group SU_q(2) and the two-dimensional quantum plane a la Woronowicz (1987) and Manin (1988), the covariant spinor calculus, very similar to the classical SU(2) group, is presented. New features arise from the noncommutation among the entries of the quantum matrix and among the 'coordinates' of the quantum plane. q-deformed Pauli matrices are defined and their applications are illustrated.