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Nonlocal diffusion equations derived from the continuous time random walk theory have been widely used in modelling the anomalous diffusion processes arising complex systems. The purpose of this book is to give an introduction to the existence and asymptotic behavior theory for diffusion equations with the general Caputo-type fractional derivative. Firstly, the unique existence of solutions of the general Caputo-type fractional ordinary differential equations is proved. Secondly, we obtain the existence and long time behavior results for the time nonlocal diffusion equations with the general Caputo-type differential operator. Thirdly, we define the general Laplacian by removing some conditions for the concept of the general fractional Laplacian and derive the time-space nonlocal diffusion equations involving the general Laplacian which can capture the tempered stable processes. Finally, the existence and asymptotic behavior results for the time-space nonlocal diffusion equations are established.