We examine distal theories and structures in the context of continuous logic, providing several equivalent definitions.

By studying the combinatorics of fuzzy VC-classes, we find continuous versions of (strong) honest definitions and distal cell decompositions.

By studying generically stable Keisler measures in continuous logic, we apply the theory of continuous distality to analytic versions of graph regularity.

We will also present some examples of distal metric structures, including dual linear continua and a continuous version of o-minimality.