  Excitons, Coulomb-bound electron-hole pairs, are the fundamental excitations
governing optoelectronic properties of semiconductors. While 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 WSe$_2$. 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 allows to retrieve fundamental exciton properties
like the binding energy and the exciton-lattice coupling and to reconstruct the
real-space excitonic wave function via Fourier transform. All quantities are in
excellent agreement with microscopic calculations. Our approach provides a full
characterization of the exciton wave function and is applicable to bright and
dark excitons in semiconducting materials, heterostructures and devices.
