In this talk, I will give some history of the subject of hyperbolic
conservation laws, and its role in the theory of partial differential
equations and applied analysis. Of course, I will also try to
describe current research and to predict future directions.

The title is also a play on words, as the theory of hyperbolic partial
differential equations is bound up with the concept of time. But
while many of the most frequently encountered linear hyperbolic
equations, such as the wave equation, are well-posed in both forward
and backward time directions, a salient feature of nonlinear
hyperbolic conservation laws is that one must break the
forward-backward time symmetry to establish a class of functions in
which the equation is well-posed. This task is often described as
"bringing in more physics", even though it can be described in
purely mathematical terms. I will describe some ways in which it has
been accomplished.