A cat is released upside down with zero initial angular momentum, and yet still manages to land on her feet. For a rigid body zero angular momentum implies zero angular velocity, but a flexible body, even with vanishing angular momentum, can rotate by changing its shape. In general, the net rotation is nonzero even if the shape returns to its initial value, so that the orientation is not a function of the shape, but rather of the shape history. Thus elementary mechanics gives rise to a kind of "geometrical phase" of rotations of flexible bodies, which in turn leads to the non-Abelian, SO(3) gauge theory of Coriolis forces in the dynamics of flexible bodies such as molecules, nuclei, stars, etc. For example, it turns out that Coriolis forces in the 3-body problem are described by a Dirac-type monopole field with charge 1/2, located at the 3-body collision. The Hamiltonian for a system of N particles, after eliminating the rotational degrees of freedom, involves the Coriolis gauge field and the metric tensor on shape space. These fields satisfy a system of field equations of the Yang-Mills-Einstein type. If time allows, the problem of small vibrations in rotating systems will be discussed, including the geometrical meaning of the Eckart conditions.