There are a number of invariants defined via Frobenius in the study of singularities in positive characteristic commutative algebra. One such is the F-signature, which can be viewed as a quantitative measure of F-regularity – an important class of singularities central to the celebrated theory of tight closure pioneered by Hochster and Huneke, and closely related to KLT singularities via standard reduction techniques from characteristic zero. After giving an overview of the theory of F-signature and its history, in this talk we’ll discuss a pair of exciting new variants of the original definition of F-signature. The first, called the dual F-signature, gives a measure of F-rationality – another important class of F-singularity closely related to rational singularities in characteristic zero. The second, called the perfectoid signature, gives an analog of the F-signature in mixed characteristics. (These two topics are based on recent joint works with Smirnov and Cai-Lee-Ma-Schwede-Tucker.)