In many languages modal auxiliaries such as English can, must, may, need, will, ought, want are ambiguous between two types of interpretations: epistemic and root interpretations. In the epistemic interpretation the modal expresses how likely it is that a proposition is true (for example, necessarily, possibly, probably true) while in the root interpretations the modal expresses the obligatoriness, permissibility, desirability, or possibility of a state or event. A central question in much syntactic research on modal auxiliaries has been whether this systematic semantic ambiguity corresponds to a syntactic distinction. A commonly accepted answer has been that in epistemic interpretations the modal verb is a monadic predicate while in root interpretations it is a dyadic predicate, typically a relation between a subject and an infinitival verb. This distinction between monadic and dyadic modal predicates has been modeled syntactically in various ways: (i) in terms of lexical argument structure, that is, as the distinction between raising and control verbs; (ii) in terms of different base positions in the array of functional heads making up the clausal spine, with epistemic modals being higher than root modals; (iii) in terms of a higher syntactic position for epistemically interpreted modals after raising at the level of semantic interpretation (LF raising); (iv) in terms of the nature of the complement of the modal. This chapter evaluates these proposals, drawing on data from, among others, English, Dutch, Icelandic, German, and Catalan and taking into account cross-linguistic differences in the modal systems. One important conclusion is that the alleged correspondence between the epistemic/root distinction and the raising/control distinction is too simple, as there are sentences with root interpretations but a raising syntax. The chapter ends with a list of questions for future research.