I will give a brief overview of a long project that started a decade ago (in collaboration with Ilia Itenberg and Sinan Sertöz and in parallel with Sławomir Rams and Matthias Schütt) and originally intended to bridge a minor gap in the proof of Segre's celebrated theorem on 64 lines on a smooth quartic surface. Confining ourselves to polarized K3-surfaces, now we manage to answer questions that no one even dared to ask, mostly because of lack of tools. For example, we
- obtained sharp upper bounds on the possible number of lines on a smooth polarized K3-surface of any degree,
- obtained similar bounds for quartics, sextics, and octics with singularities,
- advanced in the understanding of conics on K3-surfaces (sharp upper bounds for quartics, sextics, and octics),
- started the study of twisted cubics,
- made a few further steps towards the understanding of smooth rational curves on polarized K3-surfaces.
I will try to discuss both ``classical'' (that is, more than 5 years old) results and recent advances, which keep coming on a daily basis; if time permits, I will also try to outline the techniques used. | 
