In this paper we compute the real ^-groups KO*(PSp(n^) of the projective symplectic groups PSp(n). As for the complex ^-groups, we have in [6, 8] two kinds of methods of computing 7f*(G) in general for a compact connected Lie group G with finite fundamental group of prime order and by using actually those methods its ring structure is explicitly described. Neither of them deals with the projective unitary groups PU(n) except the case when n is prime. In more general, however, the case when n is a power of prime is investigated in more earlier times but not explicitly [18]. The computation in any case is based on the fact [7] that ^*(G0) (where G0 is simply connected) is an exterior algebra generated by elements of degree one arising from the basic irreducible complex representations of G0 . It seems not easy to find a comprehensive method of computing KO* (G) as in the complex case. So we proceed case by case and determined the JfO-groups of SOM, PEQ, PE7, and PSp(2 )in [10, 11, 12, 13, 14, 15]. Then we also use essentially the structure theorem on ^CO*(G0) [17] analogous to that on #*(G0). We compute KO*(PSpW) by applying the modification of the method used for the computation of X0*(PSp(2)). Making use of the equi variant ^0-theory KOz/2, especially the Thorn isomorphism theorem for KOz/2theory, we reduce the structure of KO*(PSpM) to those of KO*(SpM} and #O*(P*) for a certain integer ÂŁ > 0 where P denotes a real projective ÂŁ -space. Then we need K*(PSp(n^ with a basis in the form conforming to our method and so we begin by computing this group by a way similar to that used in computing