  Parton showers are among the most widely used tools in collider physics.
Despite their key importance, none so far has been able to demonstrate accuracy
beyond a basic level known as leading logarithmic (LL) order, with ensuing
limitations across a broad spectrum of physics applications. In this letter, we
propose criteria for showers to be considered next-to-leading logarithmic (NLL)
accurate. We then introduce new classes of shower, for final-state radiation,
that satisfy the main elements of these criteria in the widely used large-$N_C$
limit. As a proof of concept, we demonstrate these showers' agreement with
all-order analytical NLL calculations for a range of observables, something
never so far achieved for any parton shower.
