Fast, accurate computation of geophysical fluid dynamics is often very challenging. This is due to the complexity of the PDEs themselves and their initial and boundary conditions. There are several practical advantages to using a relatively new numerical method, the spectral-element method (SEM), over standard methods. SEM combines spectral-method high accuracy with the geometric flexibility and computational efficiency of finite-element methods. This paper is intended to augment the few descriptions of SEM that aim at audiences besides numerical-methods specialists. Advantages of SEM with regard to flexibility, accuracy, and efficient parallel performance are explained, including sufficient details that readers may estimate the benefit of applying SEM to their own computations. The spectral element atmosphere model (SEAM) is an application of SEM to solving the spherical shallow-water or primitive equations. SEAM simulated decaying Jovian atmospheric shallow-water turbulence up to resolution T1067, producing jets and vortices consistent with Rhines theory. SEAM validates the Held–Suarez primitive equations test case and exhibits excellent parallel performance. At T171L20, SEAM scales up to 292 million floating-point operations per second (Mflops) per processor (29% of supercomputer peak) on 32 Compaq ES40 processors (93% efficiency over using 1 processor), allocating 49 spectral elements/processor. At T533L20, SEAM scales up to 130 billion floating-point operations per second (Gflops) (8% of peak) and 9 wall clock minutes per model day on 1024 IBM POWER3 processors (48% efficiency over 16 processors), allocating 17 spectral elements per processor. Local element-mesh refinement with 300% stretching enables conformally embedding T480 within T53 resolution, inside a region containing 73% of the forcing but 6% of the area. Thereby the authors virtually reproduced a uniform-mesh T363 shallow-water computation, at 94% lower cost.