In Appendix B to their Model Theory, Chang and Keisler list some problems and conjectures that, at the time of publication, were unsolved. A few of them take imperative form, for instance:

"Develop a theory of models which stresses the order type of the model $\mathfrak{A} = \langle A, <, \dots\rangle$ rather than the cardinality of the set $A$."

Has anyone worked on this?