In this dissertation, we discuss the controllability and regularity for some control systemsgoverned by partial differential equations.We study the controllability of two classes of important quasilinear degenerate parabolicequations (Newtonian filtration equations and non-Newtonian filtration equations, whichare also called evolution p-Laplacian equations). For Newtonian filtration equation, firstly,we discuss the approximate controllability with the nonnegative constraint on the controls.When the control is on the whole domain, we characterize the set of nonnegative targetswhich are approximately controllable at any time T＞0. Moreover, we conclude that anynonnegative target is approximately controllable for a tong time. While the control is onthe local domain, we get the noncontrollability. Next, we discuss the null controllabilityof Newtonian filtration equation. We prove that for suitable initial values, system is nullcontrollable under a local control at any time T＞0. For non-Newtonian filtration equation,we also prove the null controllability.We also study the controllability of two classes of cascade systems. Firstly, we discussthe existence of insensitizing controls for a class of linear degenerate parabolic equations.As we all know, it is equivalent to the null controllability of a cascade system governedby two linear degenerate parabolic equations. Next, we study the null controllability of acascade system governed by a semilinear degenerate parabolic equation and a semilinear heatequation. Under some conditions, these systems are null controllable at any time.Finally, we study the eventual differentiability of a C_0-semigroup associated to a waveequation with boundary dissipation. Furthermore, we get that the solution has the propertyof eventual regularity and this semigroup satisfies the spectrum-determined growth condition,which is important in the stability analysis.