This chapter starts with the Lifshitz–Kosevich formula, which describes quantum oscillations and their reduction factors due to the effects of finite temperature, scattering and spin. Although quantum oscillations can generally be observed only for closed Fermi surfaces, quantum oscillations can sometimes be observed even for open Fermi surfaces, where the interband tunneling effect, called magnetic breakdown, plays an essential role. When the oscillatory part of the magnetization is sufficiently large, the waveform of the quantum oscillation is strongly deformed, which gives rise to a variety of frequencies. In contrast to quantum oscillations, the resistance can oscillate with magnetic field for open orbits without Landau quantization, which arises from quantum interference of the electron wave function. In materials with characteristic energy bands, the electrons acquire a phase as they move along an orbit on the Fermi surface. This phase is called the geometric phase or Berry phase. We will also look at how the Berry phase appears in the quantum oscillations and how it is experimentally determined.