In these four lectures on topics under this heading, we will give a panoramic view of the motivations, known techniques and methods, results, open problems, future directions, and ideas on how to proceed with the unsolved problems. The discussion will start from scratch, requiring minimal background knowledge. Plan to include the following topics in the discussion (subject to time constraints): (i) Value distribution theory and hyperbolicity, from the viewpoint of log-pole jet differentials. (ii) Deformational invariance of plurigenera and pluricanonical Hodge decomposition. (iii) Construction of rational curves in Fano manifolds by singularity-magnifying Monge-Ampere equations. (iv) Effective results in algebraic geometry and the abundance conjecture. (v) Estimates of the complex Neumann problem and the Hodge conjecture for (p,q)-type in the setting of Cauchy-Riemann geometry.