Renormalization group (RG) evolution of the neutrino mass matrix may take the value of the mixing angle θ 13 very close to zero, or make it vanish. On the other hand, starting from θ 13 = 0 at the high scale it may be possible to generate a non-zero θ 13 radiatively. In the most general scenario with non-vanishing CP violating Dirac and Majorana phases, we explore the evolution in the vicinity of θ 13 = 0, in terms of its structure in the complex U e3 plane. This allows us to explain the apparent singularity in the evolution of the Dirac CP phase δ at θ 13 = 0. We also introduce a formalism for calculating the RG evolution of neutrino parameters that uses the Jarlskog invariant and naturally avoids this singular behaviour. We find that the parameters need to be extremely fine-tuned in order to get exactly vanishing θ 13 during evolution. For the class of neutrino mass models with θ 13 = 0 at the high scale, we calculate the extent to which RG evolution can generate a nonzero θ 13 , when the low energy effective theory is the standard model or its minimal supersymmetric extension. We find correlated constraints on θ 13 , the lightest neutrino mass m 0 , the effective Majorana mass m ee measured in the neutrinoless double beta decay, and the supersymmetric parameter tan β.