We analyze random isotropic antiferromagnetic SU(N) spin chains using the real-space renormalization group. We find that they are governed at low energies by a universal infinite randomness fixed point different from the one of random spin-1/2 chains. We determine analytically the important exponents: the energy-length scale relation is Omega similar to exp(-L-psi), where psi = 1/N, and the mean correlation function is given by C-ij similar to (-1)(i-j)/\i-j\(phi), where phi = 4/N. Our analysis shows that the infinite-N limit is unable to capture the behavior obtained at any finite N.7018