The review summarizes the research carried out in the Laboratory of Nano- and Microstructural Materials at the Centre of Polymer and Carbon Materials, Polish Academy of Sciences (CMPW PAS). Studies carried out for many years under the guidance of Professor Andrzej Dworak led to the development and exploration of the mechanisms of oxirane and cyclic imine polymerization and controlled radical polymerization of methacrylate monomers. Based on that knowledge, within the last three decades, macromolecules with the desired composition, molar mass and topology were obtained and investigated. The ability to control the structure of the synthesized polymers turned out to be important, as it provided a way to tailor the physiochemical properties of the materials to their specific uses. Many linear polymers and copolymers as well as macromolecules with branched, star, dendritic and hyperbranched architectures were synthesized. Thanks to the applied controlled polymerization techniques, it was possible to obtain hydrophilic, hydrophobic, amphiphilic and stimulus-sensitive polymers. These tailor-made polymers with controlled properties were used for the construction of various types of materials, primarily on the micro- and nanoscales, with a wide range of possible applications, mainly in biomedicine. The diverse topology of polymers, and thus their properties, made it possible to obtain various types of polymeric nanostructures and use them as nanocarriers by encapsulation of biologically active substances. Additionally, polymer layers were obtained with features useful in medicine, particularly regenerative medicine and tissue engineering.