Morphologies of genus-1 and 2 toroidal vesicles are studied numerically by dynamically triangulated membrane models and experimentally by confocal laser microscopy. Our simulation results reproduce shape transformations observed in our experiments well. At large reduced volumes of the genus-1 vesicles, obtained vesicle shapes agree with the previous theoretical prediction, in which axisymmetric shapes are assumed: double-necked stomatocyte, discoidal toroid, and circular toroid. However, for small reduced volumes, it is revealed that a non-axisymmetric discoidal toroid and handled discocyte exist in thermal equilibrium in the parameter range, in which the previous theory predicts axisymmetric discoidal shapes. Polygonal toroidal vesicles and subsequent budding transitions are also found. The entropy caused by shape fluctuations slightly modifies the stability of the vesicle shapes.