Many transport processes in nature exhibit anomalous diffusive properties with nontrivial scaling of the mean square displacement, e.g., diffusion of cells or of biomolecules inside the cell nucleus, where typically a crossover between different scaling regimes appears over time. Here, we investigate a class of anomalous diffusion processes that is able to capture such complex dynamics by virtue of a general waiting time distribution. We obtain a complete characterization of such generalized anomalous processes, including their functionals and multipoint structure, using a representation in terms of a normal diffusive process plus a stochastic time change. In particular, we derive analytical closed form expressions for the two-point correlation functions, which can be readily compared with experimental data. Diffusive transport is usually classified in terms of the mean square displacement (MSD): MSDðtÞ¼h(RðtÞ−R 0 ) 2 i, where RðtÞ is a time-dependent stochastic vector, either the position or the velocity, and R 0 is its initial value. Motivated by numerous experimental results of the last decade (see Refs. [1,2] and the references therein), we usually distinguish between normal and anomalous diffusive processes for which the MSD scales linearly in time or as a power law, respectively [1][2][3][4][5][6]. However, because of the improvement of experimental techniques, evidence of more complicated nonlinear MSDs, characterized by different scaling regimes over the measurement time, has been found in recent experiments of diffusion in biophysical systems [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Here, we investigate a general class of anomalous diffusive processes that can capture such complicated MSD behavior by means of a generalized waiting time distribution. We provide a complete characterization of these processes including (i) the stochastic description of the microscopic diffusive dynamics, (ii) evolution equations for the probability density function (PDF) of the process and its associated time-integrated observables, and (iii) the multipoint correlation functions. Our general model includes as a special case the continuous time random walk (CTRW), which is widely used to model MSDs with a power-law scaling [15,18,21]. Even though CTRWs have been in the focus of theoretical research on anomalous diffusion for almost two decades, the full characterization of CTRWs in terms of (i)- (iii) has not yet been presented. In this Letter we provide the key for such a complete stochastic description by using the stochastic calculus of random time changes.The stochastic trajectory of a CTRW YðtÞ is expressed in terms of coupled Langevin equations [23]:where for convenience we focus on a process in one dimension. The CTRW is then given by YðtÞ ¼ X(SðtÞ),where the process S is defined as the inverse of T or, more precisely, as the collection of first passage times:The dynamics of X is that of a normal diffusive process in the operational time s. Thus, FðxÞ and σðxÞ satisfy standard conditions [24] an...