Two-component Fermi gas with zero-range interaction at infinite scattering length (unitary Fermi gas) is a new strongly interacting (scale-free) matter realized in cold atom experiments and has attracted intense attention across many subfields of physics. However, its analytic treatment has been difficult because of the strong interaction and hence the lack of a small expansion parameter. In this talk, we show that systematic expansions for the unitary Fermi gas are possible around d=4 and d=2 where d is the dimensionality of space. The unitary Fermi gas near d=4 can be understood as a weakly-interacting system of fermionic and bosonic degrees of freedom, while near d=2 it reduces to a weakly-interacting Fermi gas. We calculate various physical quantities, such as the thermodynamic functions, the quasiparticle spectrum, and the critical temperature, using 4-d or d-2 as a small parameter of the perturbative expansion. We argue that d=4 and d=2 are useful starting points to understand the unitary Fermi gas in reality at d=3.