This thesis consists of two independent parts. In the first part we discuss group-valued moment maps. Using group-valued implosion, introduced by Hurtubise, Jeffrey and Sjamaar, we construct a new class of examples of quasi-Hamiltonian spaces. Associated to each compact Lie group G there is a universal imploded space D(G)impl . For G = Sp(n) we show that there is a stratum of D(G)impl which has a smooth closure diffeomorphic to HPn - a quaternionic projective space. We show that HPn and S 2n exhaust all examples arising from this construction. The second part is concerned with "conjugation spaces". In particular we study conjugation spaces with a compatible Lie group action. For Lie groups of type A and C, we show that there is a degree halving ring isomorphism from equivariant cohomology of the space to equivariant cohomology of its fixed point set under an involution.