This PhD thesis consists of two main parts. The first part describes the dynamics of an ideal fluid on a stationary free surface of a given shape. It turns out that one can formulate a set of self-contained equations of momentum conservation for the tangential flow, with no reference to the flow of the fluid bulk. With these equations, one can in principle predict the surface flow on a given free surface, once its shape has been measured. The equations are expressed for a general surface using Riemannian geometry and their solutions are discussed, including some difficulties that may arise. Furthermore, the equations are applied to an experiment involving a poorly understood symmetry-breaking instability of a rotating fluid with a free surface, cf. Bergmann et al., [J. Fl. Mech. 679, 415-431 (2011)], with the result confirmed by direct measurement. This experiment is discussed in some detail together with an ongoing investigation of the fluid motion in question and the elusive instability mechanism. The second main part of the thesis describes work on point vortex dynamics and instability. The problem of point vortex pair scattering is briefly revisited together with a short discussion of chaotic advection, and the stability of vortex leapfrogging is investigated within the framework of Floquet theory. An analytical criterion is found, giving the exact location of the transition to instability earlier observed in numerical investigations by Acheson [Eur. J. Phys. 21, 269-273 (2000)]. Finally, an experimental work on elastic collisions of wet spheres is briefly discussed.