Model theory of second order logic

Suppose a model or a model class is definable in set theory. I ask, what more do we know about it, if we know that it is definable in second order logic? Second order model theory is undoubtedly manifestly different from first order model theory. It becomes quickly clear that in the case of second order logic the term “model theory" gets a new meaning. There are (some) general methods that can be used and are typical. These methods are by far not as powerful as the methods used in first order model theory. I try to isolate what is it that is characteristic to second order model theory, however little there is of it. I give a lot of space to the topic of second order characterizable stuctures and to the related topic of categoricity of complete second order theories. I point out that using so-called Henkin models, thereby restoring the Compactness Theorem and other basic principles of model theory, does not reduce second order model theory to first order model theory but rather to something between first order number theory and first order set theory.