  Laser spectroscopy of muonic atoms has been recently used to probe properties
of light nuclei with unprecedented precision. We introduce nuclear effects in
hydrogen-like atoms, nucleon structure quantities (form factors, structure
functions, polarizabilities) and their effects in the Lamb shift and hyperfine
splitting (HFS) of muonic hydrogen ($\mu$H). Updated theory predictions for the
Lamb shift and HFS in $\mu$H are presented. We review the challenges of the
ongoing effort to measure the ground-state HFS in $\mu$H and its impact on our
understanding of the nucleon spin structure. To narrow down this search, we
present a novel theory prediction obtained by scaling the measured HFS in
hydrogen leveraging radiative corrections. We also summarize recent
developments in the spectroscopy of simple atomic and molecular systems and
emphasize how they allow for precise determinations of fundamental constants,
bound-state QED tests and New Physics searches.
