We discuss the amazing properties of the unitary Fermi gas, which is the dilute and ultracold gas of fermions with infinite scattering length. Then, we illustrate the zero-temperature density functional of the unitary Fermi gas we have recently introduced on the basis of Monte Carlo calculations. This functional includes a gradient term which is essential to describe accurately the surface effects of the system, in particular with a small number of atoms. We analyze in detail the role of this gradient term for the determination of density profiles and collective modes. Finally, we investigate the low-temperature thermodynamics of the unitary Fermi gas by using the zero-temperature spectra of both bosonic collective modes and fermonic single-particle excitations. We determine the superfluid fraction, the critical temperature, the first sound and the second sound, comparing our analytical results with other theoretical predictions and experimental data.