In this paper the solutions \(u_{\nu}=u_{\nu}(x,t)\) to fractional diffusion equations of order \(0<\nu \leq 2\) are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order \(\nu =\frac{1}{2^n}\), \(n\geq 1,\) we show that the solutions \(u_{{1/2^n}}\) correspond to the distribution of the \(n\)-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order \(\nu =\frac{2}{3^n}\), \(n\geq 1,\) is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that \(u_{\nu}\) coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions \(u_{\nu}\) and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.