Every spectrum $K$ is the coefficient object of a generalized cohomology theory and dually of a generalized homology theory.
For $K = H R$ an Eilenberg-MacLane spectrum this reduces to ordinary homology
Original articles include
See also
Friedrich Bauer, Classifying spectra for generalized homology theories Annali di Maternatica pura ed applicata (IV), Vol. CLXIV (1993), pp. 365-399
Friedrich Bauer, Remarks on universal coefficient theorems for generalized homology theories Quaestiones Mathematicae Volume 9, Issue 1 & 4, 1986, Pages 29 - 54
A general construction of homologies by “geometric cycles” similar to the Baum-Douglas geometric cycles for K-homology is discussed in
Further generalization of this to bivariant cohomology theories is in