  This paper gives a leisurely introduction to Calabi-Yau manifolds and special
Lagrangian submanifolds from the differential geometric point of view, followed
by a survey of recent results on singularities of special Lagrangian
submanifolds, and their application to the SYZ Conjecture.
  It is aimed at graduate students in Geometry, String Theorists, and others
wishing to learn the subject, and is designed to be fairly self-contained. It
is based on lecture courses given at Nordfjordeid, Norway and MSRI, Berkeley in
June and July 2001.
  We introduce Calabi-Yau m-folds via holonomy groups, Kahler geometry and the
Calabi Conjecture, and special Lagrangian m-folds via calibrated geometry.
`Almost Calabi-Yau m-folds' (a generalization of Calabi-Yau m-folds useful in
special Lagrangian geometry) are explained and the deformation theory and
moduli spaces of compact special Lagrangian submanifolds in (almost) Calabi-Yau
m-folds is described.
  In the final part we consider isolated singularities of special Lagrangian
m-folds, focussing mainly on singularities locally modelled on cones, and the
expected behaviour of singularities of compact special Lagrangian m-folds in
generic (almost) Calabi-Yau m-folds. String Theory, Mirror Symmetry and the SYZ
Conjecture are briefly discussed, and some results of the author on
singularities of special Lagrangian fibrations of Calabi-Yau 3-folds are
described.
