  We discuss the deformation theory of special Lagrangian (SL) conifolds in
complex space C^m. Conifolds are a key ingredient in the compactification
problem for moduli spaces of compact SLs in Calabi-Yau manifolds. This category
allows for the simultaneous presence of conical singularities and of
non-compact, asymptotically conical, ends. Our main theorem is the natural next
step in the chain of results initiated by McLean and continued by the author
and by Joyce. We emphasize a unifying framework for studying the various cases
and discuss analogies and differences between them. This paper also lays down
the geometric foundations for our paper "Special Lagrangian conifolds, II"
concerning gluing constructions for SL conifolds in C^m.
