Building on our recent work on induced-charge electro-osmosis (ICEO) and electrophoresis (ICEP), as well as the Russian literature on spherical metal colloids, we examine the rich consequences of broken geometric and field symmetries upon the ICEO flow around conducting bodies. Through a variety of paradigmatic examples involving ideally polarizable (e.g. metal) bodies with thin double layers in weak fields, we demonstrate that spatial asymmetry generally leads to a net pumping of fluid past the body by ICEO, or, in the case of a freely suspended colloidal particle, translation and/or rotation by ICEP. We have chosen model systems that are simple enough to admit analysis, yet which contain the most important broken symmetries. Specifically, we consider (i) symmetrically shaped bodies with inhomogeneous surface properties, (ii) ‘nearly symmetric’ shapes (using a boundary perturbation scheme), (iii) highly asymmetric bodies composed of two symmetric bodies tethered together, (iv) symmetric conductors in electric-field gradients, and (v) arbitrarily shaped conductors in general non-uniform fields in two dimensions (using complex analysis). In non-uniform fields, ICEO flow and ICEP motion exist in addition to the more familiar dielectrophoretic forces and torques on the bodies (which also vary with the square of the electric field). We treat all of these problems in two and three dimensions, so our study has relevence for both colloids and microfluidics. In the colloidal context, we describe principles to ‘design’ polarizable particles which rotate to orient themselves and translate steadily in a desired direction in a DC or AC electric field. We also describe ‘ICEO spinners’ that rotate continuously in AC fields of arbitrary direction, although we show that ‘near spheres’ with small helical perturbations do not rotate, to leading order in the shape perturbation. In the microfluidic context, strong and steady flows can be driven by small AC potentials applied to systems containing asymmetric structures, which holds promise for portable or implantable self-powered devices. These results build upon and generalize recent studies in AC electro-osmosis (ACEO). Unlike ACEO, however, the inducing surfaces in ICEO can be physically distinct from the driving electrodes, increasing the frequency range and geometries available.