We consider a fractional version of the classical nonlinear birth process of which the Yule–Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the difference-differential equations which govern the probability law of the process with the Dzherbashyan–Caputo fractional derivative. We derive the probability distribution of the number $\mathcal{N}_{\nu}(t)$ of individuals at an arbitrary time t. We also present an interesting representation for the number of individuals at time t, in the form of the subordination relation $\mathcal{N}_{\nu}(t)=\mathcal{N}(T_{2\nu}(t))$ , where $\mathcal{N}(t)$ is the classical generalized birth process and T2ν(t) is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed.