Sustained floating-point computation rates on real applications, as tracked by the ACM Gordon Bell Prize, increased by three orders of magnitude from 1988 (1 Gigaflop/s) to 1998 (1 Teraflop/s), and by another three orders of magnitude to 2008 (1 Petaflop/s). Computer engineering provided only a couple of orders of magnitude of improvement for individual cores over that period; the remaining factor came from concurrency, which is approaching one million-fold. Algorithmic improvements contributed meanwhile to making each flop more valuable scientifically. As the semiconductor industry now slips relative to its own roadmap for silicon-based logic and memory, concurrency, especially on-chip many-core concurrency and GPGPU SIMD-type concurrency, will play an increasing role in the next few orders of magnitude, to arrive at the ambitious target of 1 Exaflop/s, extrapolated for 2018. An important question is whether today's best algorithms are efficiently hosted on such hardware and how much co-design of algorithms and architecture will be required. From the applications perspective, we illustrate eight reasons why today's computational scientists have an insatiable appetite for such performance: resolution, fidelity, dimension, artificial boundaries, parameter inversion, optimal control, uncertainty quantification, and the statistics of ensembles. The paths to the exascale summit are debated, but all are narrow and treacherous, constrained by fundamental laws of physics, cost, power consumption, programmability, and reliability. Drawing on recent reports, workshops, vendor projections, and experiences with scientific codes on contemporary platforms, we propose roles for today's researchers in one of the great global scientific quests of the next decade.