After 15-20 years of architectural stability, we are in the midst of a dramatic change in high performance computing systems design. In this talk we discuss the commonalities across the viable systems of today, and look at opportunities for numerical algorithms research and development. In particular, we explore possible programming and machine abstractions and how we can develop effective algorithms based on these abstractions, addressing, among other things, robustness issues for preconditioned iterative methods and resilience of algorithms in the presence of soft errors.