Excitons, Coulomb-bound electron-hole pairs, are the fundamental excitations governing the optoelectronic properties of semiconductors. Although optical signatures of excitons have been studied extensively, experimental access to the excitonic wave function itself has been elusive. Using multidimensional photoemission spectroscopy, we present a momentum-, energy-, and time-resolved perspective on excitons in the layered semiconductor WSe2. By tuning the excitation wavelength, we determine the energy-momentum signature of bright exciton formation and its difference from conventional single-particle excited states. The multidimensional data allow to retrieve fundamental exciton properties like the binding energy and the exciton-lattice coupling and to reconstruct the real-space excitonic distribution function via Fourier transform. All quantities are in excellent agreement with microscopic calculations. Our approach provides a full characterization of the exciton properties and is applicable to bright and dark excitons in semiconducting materials, heterostructures, and devices.|Key points:|center dot The full life cycle of excitons is recorded with time- and angle-resolved photoemission spectroscopy.|center dot The real-space distribution of the excitonic wave function is visualized.|center dot Direct measurement of the exciton-phonon interaction.