  We obtain a rigorous solution of universal Bose gases near resonance and
offer an answer to one of the long-standing challenges of quantum gases at
large scattering lengths, where the standard dilute theory breaks down. The
solution was obtained by using an $\epsilon$ expansion near four spatial
dimension. In dimension $d = 4 - \epsilon$, the chemical potential of Bose
gases near resonances is shown to approach the universal value
$\epsilon^{(2/(4-\epsilon))} \epsilon_F \sqrt{2/3} (1 + 0.474 \epsilon - i
1.217 \epsilon + ...)$, where $\epsilon_F$ is the Fermi energy defined for a
Fermi gas of density $n$, and the condensation fraction is equal to $2/3 (1 +
0.0877 \epsilon + ...)$. We also discuss the implications on ultra-cold gases
in physical dimensions.
