There was a huge progress in understanding the geometric nature of removable singularities for some subtle classes of analytic and harmonic functions. This progress culminated in the solution of several problems of Vitushkin from the 50's. A 100 year old problem of Painlevé asking for ``geometric" description of removable sets for bounded analytic functions was solved by Xavier Tolsa. The progress was based on essentially several ingredients including: 1. very non-trivial connection between Geometric Measure Theory and Calderon-Zygmund operators and 2. a theory of Calderon-Zygmund operators in ``bad environment". The talk will present this story, which unites the methods of Complex Analysis, Harmonic Analysis, Geometric Measure Theory and Probability. Some unsolved problems will be listed.