Atoms trapped in optical lattices have been used successfully to study many-body phenomena. However, the shape that bosonic ground-state wavefunctions can take is limited, compromising the usefulness of this approach. Such limitations, however, do not apply to excited states of bosons. The study of atomic superfluids realized in higher-energy bands, where orbital degrees of freedom are essential, can bring the world of optical lattices closer to relevant condensed matter systems or allow to study scenarios not at all realized in nature. I will discuss our observations of unconventional superfluid order parameters in the 2nd and 7th band, and the observation of a topologically induced avoided crossing between the 2nd, 3rd and 4th bands.