Estimating the yield of underground nuclear explosions from the far-field seismic waves is a difficult problem, since accurate estimates require a detailed accounting for source-region and travel-path differences between known and unknown yield events. For short-period P waves, the first problem is to obtain an unbiased measure of the far-field energy. Conventional mb seem unsatisfactory because of the large scatter in station values. The dominant cause seems to be wave focusing/defocusing by complex earth structure. Improved routine measures are likely to involve measurements of the spectral energy in the initial P and/or P coda. Direct modeling of recordings from especially good stations can also provide important information. Even with an unbiased measure of far-field energy, bias in yield estimates can result from regionally varying path effects (mainly attenuation) and variations in the source-coupling properties of the local geology. The effect of attenuation differences is difficult to quantify for specific events due to strong tradeoff with source-coupling and complex earth structure effects. Detailed computer models for the effect of material properties on coupling are able to match most of the available data, but their validity for extrapolation to other materials is questioned because of problems with uniqueness and independent verification of the model parameters. The NTS volcanic materials are best understood. The major unresolved problems are with granite and salt, where the bulk material strength at the large scale of nuclear explosions controls the coupling and is very difficult to estimate. For surface waves, the issues are analogous, except for the additional complication of nonisotropic effects that sometimes dominate the radiation. Otherwise, it appears that surface waves should give more precise estimates for large events. Yield estimation is, in practice, done with some empirical relationship based on known yield events. This is briefly discussed to indicate that, despite the many technical problems, mb and Mb data scale rather well with yield. The main objective of the paper is to summarize the current state of understanding of the technical issues and highlight the important questions.