  Excitons -- two-particle correlated electron-hole pairs -- are the dominant
low-energy optical excitation in the broad class of semiconductor materials,
which range from classical silicon to perovskites, and from two-dimensional to
organic materials. Recently, the study of excitons has been brought on a new
level of detail by the application of photoemission momentum microscopy -- a
technique that has dramatically extended the experimental capabilities of time-
and angle-resolved photoemission spectroscopy (trARPES). Here, we review how
the energy- and momentum-resolved photoelectron detection scheme enables direct
access to the energy landscape of bright and dark excitons, and, more
generally, to the momentum-coordinate of the exciton that is fundamental to its
wavefunction. Focusing on two-dimensional materials and organic semiconductors
as two tuneable platforms for exciton physics, we first discuss the typical
photoemission fingerprint of excitons in momentum microscopy and highlight that
is is possible to obtain information not only on the electron- but also
hole-component of the former exciton. Second, we focus on the recent
application of photoemission orbital tomography to such excitons, and discuss
how this provides a unique access to the real-space properties of the exciton
wavefunction. Throughout the review, we detail how studies performed on
two-dimensional transition metal dichalcogenides and organic semiconductors
lead to very similar conclusions, and, in this manner, highlight the strength
of time-resolved momentum microscopy for the study of optical excitations in
semiconductors.
