On Cepheid distances in the $H_0$ measurement

Classical Cepheids were the first stellar standard candles and have played a crucial role for astronomical distance measurements ever since the discovery of the Leavitt law (period-luminosity relation). Enormous improvements in distance accuracy have been achieved since Hertzsprung's first application of Leavitt's law to measure the distance to the Small Magellanic Cloud in 1913, notably in very recent years thanks to a large data set of highly accurate space astrometry from the ESA mission Gaia. Complemented by homogeneous space photometry, Cepheids enable the most accurate distance estimates to galaxies hosting type-Ia supernovae up to approximately 70 Mpc distant. Here, I review the history of Cepheid distance measurements, open questions on the side of stellar astrophysics, and recent studies seeking to quantify and mitigate systematics with a view to further improve the accuracy on the Hubble constant. For example, the recently launched James Webb Space Telescope will enhance precision due to 4x lower sensitivity to source blending in crowded regions and greater sensitivity in dust-insensitive infrared bands. Future 30m-class telescopes could in principle further improve Cepheid distance measurements towards the Hubble flow, if technical challenges related to a continuously evolving instrument can be overcome.