The object of the present article is to develop a topological K-theory for JC-algebras - uniformly closed Jordan algebras of bounded selfadjoint operators on a complex Hilbert space. In § 1 the authors define JC- algebra K-groups which are not isomorphism invariant and dependent on representations of the JC-algebra as operator algebra. This technical difficulty can be explained by the problem of finding a suitable substitute for matrix Banach algebras. The picture becomes clearer in § 2 where “universal” K-groups are encountered which give rise to functors from the category of JC-algebras and Jordan homomorphisms to the category of abelian groups and group homomorphisms, preserve direct limits and give desirable exact sequences. The relationship between the universal K-groups and those given in § 1 is described by an exact sequence, more detailed information on which is given in § 3 via analysis of JC-algebras versus real \(C^*\)-algebras they generate. In § 4 a connection with complex \(C^*\)-algebras K-theory is explained and for some JC-algebras K-groups are evaluated.