  This is the first in a series of five papers math.DG/0211295,
math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian
submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities
x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m
with isolated singularities at 0. Readers are advised to begin with the final
paper math.DG/0303272, which surveys the series, gives examples, and applies
the results to prove some conjectures.
  This first paper lays the foundations for the series, giving definitions and
proving auxiliary results in symplectic geometry and asymptotic analysis that
will be needed later. We also prove results on the regularity of X near its
singular points.
  We show that X converges to the cone C_i near x_i with all its derivatives,
at rates determined by the eigenvalues of the Laplacian on the intersection of
C_i with the unit sphere. We show that if X is a special Lagrangian integral
current with a tangent cone C at x satisfying some conditions, then X has an
isolated conical singularity at x in our sense. We also prove analogues of many
of our results for Asymptotically Conical SL m-folds in C^m.
  The sequel math.DG/0211295 studies the deformation theory of compact SL
m-folds X in M with conical singularities. The third and fourth papers
math.DG/0302355, math.DG/0302356 construct desingularizations of X, realizing X
as a limit of a family N^t of compact, nonsingular SL m-folds in M.
