Communication -- the movement of data between levels of memory hierarchy or between processors over a network -- is the most expensive operation in terms of both time and energy at all scales of computing. Achieving scalable performance in terms of time and energy thus requires a dramatic shift in the field of algorithmic design. Solvers for sparse linear algebra problems, ubiquitous throughout scientific codes, are often the bottlenecks in application performance due to a low computation/communication ratio. In this paper we develop three potential implementations of communication-avoiding Lanczos bidiagonalization algorithms and discuss their different computational requirements. Based on these new algorithms, we also show how to obtain a communication-avoiding LSQR least squares solver.