We describe and compare two approaches to solid subpixel target detection in hyperspectral imagery. The first approach requires explicit models for both the target and the background, and employs a generalized likelihood ratio in order to obtain a detector that is optimized to those specific models. When this approach is most successful, a closed-form solution is obtained that permits the detector to be efficiently applied. A specific example of this approach is outlined in some detail, leading to the elliptically-contoured finite-target matched filter (EC-FTMF), a variant of the classical FTMF algorithm that uses a multivariate t-distribution instead of a Gaussian as the model for the background. The second approach also requires an explicit model of the target, but does not need a model for the background. In this second approach, matched pairs of data samples are created: for each pixel in the original hyperspectral image, a corresponding pixel is generated by implanting the target into the original pixel. These matched pairs are used as training data for a machine learning algorithm to classify pixels as either non-target or target. Here we use a support vector machine, but the matched pair machine learning (MPML) framework does not restrict the choice of classifier type. Detectors using both approaches are applied both to simulated data (with Gaussian and with multivariate fit distributed backgrounds) and to real hyperspectral data with known, referenced targets.