This is a survey on results and in particular on recent development on the question: which compact subsets of the complex plane are removable for bounded analytic functions. We begin with classical results and here knowledge of only basic complex analysis is required. Later on some geometric measure theory and in particular very advanced theory of singular integrals enter into play. But since in this part we are not giving any detailed proofs, also a reader who is not familiar with these topics should be able to get an idea what is going on. In this first parts of this paper I give several proofs for many rather easy and well-known results. I thus hope that a reader who is not familiar with the subject could gain some insight into it. In the later part parts on more recent results some ideas of the proofs are only sketched. The complete details get often very complicated and are best studied in the original papers. Recent lecture notes of Pajot [P1] cover this topic in much greater detail, also in historical perspective. There one can find many more references than we give here for the background and for further reading. Other recent surveys on this and related topics are [D4], [Me2], [P2], [V1] and [V2]. Much of the background material can be found in [G], [D2], [C1], [M2] and [Mu].