This thesis has two main parts. The first part, which consists of two papers, is concerned with the role of equivariant loop spaces in the K-theory of exact categories with duality. We prove a group completion-type result for topological monoids with anti-involution. The methods in this proof also apply in the context of K-theory and we obtain a similar result there. We go on to prove equivairant delooping results for Hesselholt and Madsen's Real algebraic K-theory. From these we obtain an equivalence of the fixed points of Real algebraic K-theory with Schlichting's Grothendieck-Witt space. This equivalence implies a group completion result for Grothendieck-Witt-theory, and for Real algebraic K-theory it implies that the analogs of the Conalty and Devissage theorems hold.The second part of the thesis, which consists of one paper, is about the equivariant homotopy theory of so-called G-diagrams. Here G is a finite group that acts on a small category I. A G-diagram in a category G is a functor from I to G together with natural transformations that give a "generalized G-action" on the functor. We give a model structure on the category of I-indexed G-diagrams in C , when the latter is a suciently nice model category. Important examples are the categories of topological spaces, simplicial sets and orthogonal spectra with the usual model structures. We formulate a theory of G-linear homotopy functors in terms of cubical G-diagrams. We obtain a new proof of the classical Wirthmuller isomorphismtheorem using the fact that the identity functor on orthogonal spectra is G-linear.