The boson star equation $\sqrt{-\Delta} u - (|x|^{-1} * |u|^2) u = -u$ in $R3$ involves both a non-local kinetic and potential energy and is $L^2$-critical. We establish uniqueness, radial symmetry (up to translations) and analyticity of non-negative solutions. We also prove the nondegeneracy of the linearization. Our proof of uniqueness blends variational arguments with the harmonic extension, and our proof of radial symmetry extends the moving planes method to our non-local setting. This is joint work with E. Lenzmann.