  In this chapter we provide an introduction to fractional dissipative partial
differential equations (PDEs) with a focus on trying to understand their
dynamics. The class of PDEs we focus on are reaction-diffusion equations but we
also provide an outlook on closely related classes of PDEs. To simplify the
exposition, we only discuss the cases of fractional time derivatives and
fractional space derivatives in the PDE separately. As our main tools, we
describe analytical as well as numerical methods, which are generically
necessary to study nonlinear dynamics. We start with the analytical study of
steady states and local linear stability for fractional time derivatives. Then
we extend this view to a global perspective and consider time-fractional PDEs
and gradient flows. Next, we continue to steady states, linear stability
analysis and bifurcations for space-fractional PDEs. As a final analytical
consideration we discuss existence and stability of traveling waves for
space-fractional PDEs. In the last parts, we provide numerical discretization
schemes for fractional (dissipative) PDEs and we utilize these techniques
within numerical continuation in applied examples of fractional
reaction-diffusion PDEs. We conclude with a brief summary and outlook on open
questions in the field.
