  Cepheids have long been used as standard candles to determine distances
around the Milky Way and to nearby galaxies. A discrepancy still remains for
Hubble Constant determinations using Cepheids vs. the cosmic microwave
background or calibrations to the tip of the red-giant branch. Therefore,
refinement of Cepheid period-luminosity relations continues to be an active
topic of research.
  Cepheids are also important laboratories for testing stellar physics. This
paper explores outstanding questions in Cepheid evolution and pulsation
modeling. We examine the discrepancy between Cepheid masses determined from
pulsation properties and binary orbital dynamics and those determined using
stellar evolution models. We review attempts to resolve the discrepancy by
including rotation, convective overshooting, and mass loss. We review the
impact of uncertainties in nuclear reaction rates on Cepheid evolution and the
extent of blue loops in the Hertzsprung-Russell diagram. We consider
implications for Cepheids of stellar opacity revisions suggested in light of
findings for the Sun and other types of variable stars.
  We apply the 1-D open-source MESA stellar evolution code and the MESA radial
stellar pulsation (RSP) nonlinear hydrodynamics code to investigate changes in
input physics for Cepheid models. We touch on progress in 2-D and 3-D stellar
modeling applied to Cepheids. Additional areas in which Cepheid models are
being tested against observations include: predicting the edges of the Cepheid
pulsation instability strip; predicting period-change rates and implications
for instability strip crossings; explaining period and amplitude modulations
and periodicities that may be non-radial pulsation modes; discovering what can
be learned from Cepheid observations in X-ray, ultraviolet, and radio
wavelengths. We also show a few examples of Cepheid light curves from NASA TESS
photometry.
