We explore unconventional ways how ultracold fermions pair and form collective quantum phases exhibiting long-range order, such as superfluidity and magnetically order. Specifically, we plan to realize and study pairing with orbital angular momentum and pairing induced by long-range interaction. Besides the fundamental interest in unravelling unconventional pairing mechanisms and the interplay between superfluidity and quantum magnetism, our project will also lead to gaining experimental control over topologically protected quantum states. This will pave the way for future topological quantum computers, which are particularly robust to environmental decoherence.Our project addresses three different aspects: (1) We plan to realize p-wave superfluids in two dimensions. This quantum phase exhibits topological excitations (vortices) with anyonic statistics and an isomorphism to the fractional quantum-Hall effect. We will investigate the unusual properties of p-wave superfluids, such as Majorana fermions, i.e. quasiparticles being their own anti-particles, which are predicted to be localized at vortices. This will boost the long-standing efforts in the cold atoms and condensed matter communities to understand topological states of matter. (2) We aim to realize d-wave pairing in optical lattices using a novel experimental approach. d-wave pairing is closely related to high-Tc superconductivity in the cuprates and we are interested in exploring its interplay with magnetic order. Superfluidity and magnetic order are antagonistic phenomena from a conventional BCS-theory point-of-view and hence several fundamental questions will be answered. (3) We plan to induce long-range interactions using a high-finesse optical cavity leading to a light-induced pairing mechanism. We will search for Cooper pairing in spin-polarized Fermi gases mediated by the interaction of Fermions with a quantized light field. This provides access to a new class of combined light-matter quantum states.