In the next years we expect a dramatic leap in our knowledge of high-energy astrophysical phenomena, triggered by the first observatory for very-high-energy gamma-ray astronomy, the Cherenkov Telescope Array (CTA). Thanks to the unprecedented sensitivity (more than one order of magnitude better than current instruments), the improved angular resolution and the wide energy coverage (from 20 GeV up to 300 TeV), CTA carries an enormous scientific potential for discovery. After an introduction on the detection technique and the expected performances, I will review some of the more interesting physical topics in the focus of CTA. In particular I will discuss the possibility to discover the most extreme galactic accelerators (so-called Pevatron), to study extreme phenomena acting in relativistic jets and to characterize the extragalactic background light and the intergalactic magnetic field.