I will discuss the excitation spectrum of an ultracold gas of paired fermions in its superfluid phase. There remain important open questions on this fundamental subject of many-body quantum physics, and cold atom experiments have now the capacity to answer them. Like in other superfluids, the lowest-energy excitations are phonons: collective modes of density fluctuations. But unlike most superfluids, the phonon dispersion may become subsonic in a Fermi gas, which has important consequences on the interaction mechanisms between phonons, and thus on the system's dissipative properties. At higher energy, I will show that another collective branch, related to excitations of the order-parameter amplitude, exists inside the continuum of fermionic pair-breaking excitations. The observability and dispersion relation of this branch (sometimes called ``Higgs'' branch) is still controversial in the theoretical literature while several experimental results are available. I will explain that to distinguish clearly the collective mode contribution from the continuum, one should perform an analytic continuation of the order-parameter Green's function through the branch cut associated to the continuum.Finally, in these paired systems, there exists a gapped spectrum of fermionic quasiparticles. I will explain how to described this fermionic branch beyond the mean-field approximation by viewing the quasiparticles as a polaronic impurities embedded in a sea of phonons.