Sums-of-exponentials models are widely used in biomedical research, chiefly as models of data, despite a sizable folklore criticizing their usefulness. Problems in multiexponential model fitting are addressed here, along with an exposition of how to quantify them and critically assess their quality with available statistical methods and computer programs. This class of models also is reconciled with two classes of models of systems: multicompartmental and noncompartmental models. Key issues include the importance of choosing a correct data error model, the necessity for computing model precision estimates, and the distinction between problems due to experiment design or overparameterization and purported difficulties with multiexponential models. Methods for obtaining statistical estimates of model precision, for checking goodness of fit of competing models, and for improving sampling designs are presented. Also the classic Lanczos problem is revisited, and some difficulties are resolved with a more efficient experiment design.