Harnack inequalities and applications for stochastic equations In this thesis we mainly study Wangs Harnack inequalities and their applications for the transition semigroups associated with stochastic equations. Among the stochastic equations we consider finite dimensional stochastic differential equations with irregular drifts; infinite dimensional (semi )linear stochastic partial differential equations with Gaussian or Lévy noise; multivalued stochastic differential equations in finite dimension and multivalued stochastic evolution equations in Banach spaces. The applications of Harnack inequalities include the study of regularizing properties like strong Feller property heat kernel estimates and hyperboundedness etc. for the transition semigroups. The main method we used to establish Harnack inequalities is measure transformation. There are two aspects: image measure transformation and Girsanov transformation (combined with a coupling argument). The latter method was introduced by Arnaudon et al. in 2006. By simple image measure transformation (e.g. use the Cameron Martin formula in Gaussian case) we prove an optimal Harnack inequality for the Gaussian Ornstein Uhlenbeck semigroup. This is an improvement of the Harnack inequality obtained by Röckner and Wang in 2003. Moreover we prove that this inequality is equivalent with the strong Feller property of the Gaussian Ornstein Uhlenbeck semigroup. By coupling and Girsanov transformation we obtain a Harnack inequality for Lévy driven Ornstein Uhlenbeck processes. The drift in the Girsanov transformation is a null control of some linear system. By optimizing over all null controls we obtain a Harnack inequality which is of the same form as the Harnack inequality for Ornstein Uhlenbeck processes driven by Wiener processes. This Harnack inequality also generalizes and improves the one obtained by Röckner and Wang in 2003 ...