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Cepheids have long been used as standard candles to determine distancesaround the Milky Way and to nearby galaxies. A discrepancy still remains forHubble Constant determinations using Cepheids vs. the cosmic microwavebackground or calibrations to the tip of the red-giant branch. Therefore,refinement of Cepheid period-luminosity relations continues to be an activetopic of research.  Cepheids are also important laboratories for testing stellar physics. Thispaper explores outstanding questions in Cepheid evolution and pulsationmodeling. We examine the discrepancy between Cepheid masses determined frompulsation properties and binary orbital dynamics and those determined usingstellar evolution models. We review attempts to resolve the discrepancy byincluding rotation, convective overshooting, and mass loss. We review theimpact of uncertainties in nuclear reaction rates on Cepheid evolution and theextent of blue loops in the Hertzsprung-Russell diagram. We considerimplications for Cepheids of stellar opacity revisions suggested in light offindings for the Sun and other types of variable stars.  We apply the 1-D open-source MESA stellar evolution code and the MESA radialstellar pulsation (RSP) nonlinear hydrodynamics code to investigate changes ininput physics for Cepheid models. We touch on progress in 2-D and 3-D stellarmodeling applied to Cepheids. Additional areas in which Cepheid models arebeing tested against observations include: predicting the edges of the Cepheidpulsation instability strip; predicting period-change rates and implicationsfor instability strip crossings; explaining period and amplitude modulationsand periodicities that may be non-radial pulsation modes; discovering what canbe learned from Cepheid observations in X-ray, ultraviolet, and radiowavelengths. We also show a few examples of Cepheid light curves from NASA TESSphotometry.

Challenges in Cepheid Evolution and Pulsation Modeling