We know of both planets orbiting other stars - exoplanets - and stars that themselves have stellar companions, whether it be binaries or higher-order multiples. It is natural to connect the two concepts and consider the existence of planets in multi-star systems. In this thesis we analyse the unique dynamical and observational signature of planets orbiting around a central binary, known as circumbinary planets. Particular attention is paid to planets which transit in front of their host stars. The work in this thesis is used to interpret existing discoveries, such that we better understand the abundance, distribution and formation history of circumbinary planets. We also propose and commence new complementary programs for discovering circumbinary planets, using radial velocities and astrometry. Finally, the focus is sometimes shifted away from the planets and onto the binaries themselves, aiming to solve old problems in the formation and evolution of tight stellar pairs.