Stability and instability in inertial particle dynamics The dynamics of inertial (i.e., finite-size) particles in fluid flows may differ significantly from infinitesimal fluid particle dynamics. Inertial particles turn out to be attracted to a lower-dimensional slow manifold in phase space on which the equations of motion are dissipative. In certain flow regions, the slow manifold becomes unstable and leads to an unexpected departure of inertial particle motion from infinitesimal fluid motion. Here I discuss exact analytic results for the inertial slow manifold and its instabilities. I also show applications to atmospheric contamination problems, animal locomotion, and rigid body oscillation problems. be obtained from first principles. Beyond discussing the mathematics behind the new theory, I show numerical