We report observations of toroidal vesicles with circular and noncircular cross sections, axisymmetric and nonaxisymmetric. Shape transformations induced by temperature changes are also described, which permit a connection to a mathematical conjecture due to Willmore. Our observations are analysed using a numerical procedure which allows a determination of the relevant geometrical parameters of the shapes. We compare these observations with recent theoretical predictions and point out some unexpected properties of the observed equilibrium shapes