  This is a survey of the author's series of three papers math.DG/0111324,
math.DG/0111326, math.DG/0204343 using analysis to investigate special
Lagrangian 3-folds (SL 3-folds) in C^3 invariant under the U(1)-action
(z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, and their
sequel math.DG/0011179 on special Lagrangian fibrations and the SYZ Conjecture.
  We briefly present the main results of these four long papers, giving some
explanation and motivation, but no proofs. The aim is to make the results and
ideas accessible to String Theorists and others who have an interest in special
Lagrangian 3-folds and fibrations, but have no desire to read pages of
technical analysis.
  Let N be an SL 3-fold in C^3 invariant under the U(1)-action above. Then
|z_1|^2-|z_2|^2=2a on N for some real number a. Locally, N can be written as a
kind of graph of functions u,v : R^2 --> R satisfying a nonlinear
Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic
function of x+iy. When a=0 the equations may have singular points where u,v are
not differentiable, which leads to analytic difficulties.
  We prove existence and uniqueness results for solutions u,v on domains S in
R^2 with boundary conditions, including singular solutions. We study their
singularities, giving a rough classification by multiplicity and type. We prove
the existence of large families of fibrations of open subsets of C^3 by
U(1)-invariant SL 3-folds, including singular fibres. Finally, we use these
fibrations as local models to draw conclusions about the SYZ Conjecture on
Mirror Symmetry of Calabi-Yau 3-folds.
