This article uses Cartan-Kähler theory to show that a
small neighborhood of a point in any surface with a Riemannian
metric possesses an isometric Lagrangian immersion into the
complex plane (or by the same argument, into any Kähler
surface). In fact, such immersions depend on two functions of
a single variable. On the other hand, explicit examples are
given of Riemannian three-manifolds which admit no local
isometric Lagrangian immersions into complex three-space. It
is expected that isometric Lagrangian immersions of
higher-dimensional Riemannian manifolds will almost always be
unique. However, there is a plentiful supply of flat
Lagrangian submanifolds of any complex $n$-space; we show that
locally these depend on $\frac{1}{2}n(n+1)$ functions of a
single variable.