These lecture notes start with L. Carleson's deep results on interpolation and corona problems in the unit disk and continues with modern analogues in the disk and ball. The reader learns several techniques that provide different proofs of the corona problem. These techniques from classical analysis and operator theory include duality, the Blaschke product, Hilbert space arguments, BMO, best approximations, the Beurling transform, use of trees, the complete Pick property and the Toeplitz corona theorem. The reader is assumed to know basic real and complex analysis and also the theory of the Poisson integral in the unit disk. The book contains a nice and detailed appendix on background material in functional analysis, Sobolev spaces and function theory on the disk.