Abstract. We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups P n ðRP 2 Þ of the projective plane. The maximal finite subgroups of P n ðRP 2 Þ are isomorphic to the quaternion group of order 8 if n ¼ 3, and to Z 4 if n d 4. Further, for all n d 3, the following groups are, up to isomorphism, the infinite virtually cyclic subgroups of P n ðRP 2 Þ: Z, Z 2 Â Z and the amalgamated product Z 4 Ã Z2 Z 4 .