We survey some recent developments in the study of
the complexity of lattice problems. After a discussion of some
problems on lattices which can be algorithmically solved
efficiently, our main focus is the recent progress on complexity
results of intractability. We will discuss Ajtai's worst-case/
average-case connections, NP-hardness and non-NP-hardness,
transference theorems between primal and dual lattices, and the
Ajtai-Dwork cryptosystem.

(This is a survey article appearing in the Conference on
Computational Complexity at FCRC, May 1999.)