Abstract: Unlike one-parameter persistence modules, for which we have the barcode, persistence modules with two or more parameters do not admit a complete discrete invariant, and thus incomplete invariants must be used to study the structure of such modules in practice. The multigraded Betti numbers — a standard invariant from commutative algebra — are among the simplest such incomplete invariants, and, although this invariant is already being used in applications, it is a priori unclear whether it satisfies a stability result analogous to the stability of the one-parameter barcode. Stability results are essential for the interpretability and consistency of practical methods. I will present joint work with Steve Oudot in which we prove stability results for multigraded Betti numbers. I will also discuss ongoing work in which we prove the stability of finer invariants coming from various exact structures on the category of representations of a poset. I will motivate the use of persistence from the point of view of data science and no prior knowledge of persistence theory will be assumed.