The stability of matter for nuclei and electrons interacting through classical electromagnetic fields holds only under certain conditions on the fine structure constant and the nuclear charges. The reason that instability can occur is due to the existence of zero modes for the magnetic kinetic energy operator, the Pauli operator. The Pauli operator is the square of the 3-dimensional Dirac operator and includes the interaction of the spin with the magnetic field. The zero modes of the Pauli operator are thus the same as zero modes of the Dirac opeator. The existence of zero modes for finite energy magnetic fields was first proved by Loss and Yau in 1986. Later Erdös and I found a more geometric construction of a larger class of magnetic fields leading to zero modes. A more flexible class of examples is important to better understand the limitations they cause on stability. I will explain how zero modes can also be understood through the topological concept of spectral flow for one-parameter families of Dirac operators. This is based on joint work with J. Sok And F. Portmann. Finally, I will suggest an Atiyah-Singer approach to the existence of zero modes which unfortunatey is still a conjecture.