We investigate the theories of the \emph{$\lambda Y$-calculus}, \emph{i.e.} simply typed $\lambda$-calculus with \emph{fixpoint} combinators. Non-terminating $\lambda Y$-terms exhibit a rich behavior, and one can reflect in $\lambda Y$ many results of untyped $\lambda$-calculus concerning theories. All  theories  can be characterized as \emph{contextual theories} \`a la Morris, w.r.t. a suitable set of \emph{observables}. We focus on  theories arising from natural classes of observables, where $Y$ can be approximated, albeit not always initially. In particular, we present the standard theory, induced by \emph{terminating terms}, which features a canonical interpretation of $Y$ as ``minimal fixpoint'', and another theory, induced by \emph{pure} $\lambda$-terms, which features a non-canonical interpretation of $Y$. The interest of these two theories is that the term model of the $\lambda Y$-calculus w.r.t. the first theory gives a \emph{fully complete model} of the maximal theory of the simply typed $\lambda$-calculus, while the term model of the latter theory provides a \emph{fully complete model} for the observational equivalence in unary PCF. Throughout the paper we raise open questions and conjectures.