  About 25 years ago, it came to light that a single combinatorial property
determines both an important dividing line in model theory (NIP) and machine
learning (PAC-learnability). The following years saw a fruitful exchange of
ideas between PAC learning and the model theory of NIP structures. In this
article, we point out a new and similar connection between model theory and
machine learning, this time developing a correspondence between
\emph{stability} and learnability in various settings of \emph{online
learning.} In particular, this gives many new examples of mathematically
interesting classes which are learnable in the online setting.
