Cepheids have been the cornerstone of the extragalactic distance scale for a century. With high-quality data, these luminous supergiants exhibit a small dispersion in their Leavitt (period–luminosity) relation, particularly at longer wavelengths, and few methods rival the precision possible with Cepheid distances. In these proceedings, we present an overview of major observational programs pertaining to the Cepheid extragalactic distance scale, its progress and remaining challenges. In addition, we present preliminary new results on Cepheids from the James Webb Space Telescope (JWST). The launch of JWST has opened a new chapter in the measurement of extragalactic distances and the Hubble constant. JWST offers a resolution three times that of the Hubble Space Telescope (HST) with nearly 10 times the sensitivity. It has been suggested that the discrepancy in the value of the Hubble constant based on Cepheids compared to that inferred from measurements of the cosmic microwave background requires new and additional physics beyond the standard cosmological model. JWST observations will be critical in reducing remaining systematics in the Cepheid measurements and for confirming if new physics is indeed required. Early JWST data for the galaxy, NGC 7250 show a decrease in scatter in the Cepheid Leavitt law by a factor of two relative to existing HST data and demonstrate that crowding/blending effects are a significant issue in a galaxy as close as 20 Mpc.

Water vapor maser emission from a Seyfert 2 galaxy IC2560 was observed with sensitive VLBI. The accurate black hole mass was determined by separating the mass of the disk surrounding the black hole. In addition, the distance of the galaxy was directly measured from the maser disk. The Hubble constant determined using the distances and recession velocities of IC2560 and othere megamasers is inconsistent with that determined with CMB by 3–6 km s-1 Mpc-1.

Time delay cosmography is based on the study of multiply gravitationally lensed images of a variable source. Their time delay differences are linearly dependent on the Fermat potential differences at the images’ positions and the Hubble parameter, whose exact value is to this date strongly debated in the framework of the Hubble tension. In this paper we present the study of SDSSJ1433, a quadruply lensed QSO, the time delays of which have been obtained after a 3-year observational campaign from the 2.1m Wendelstein telescope in the optical ǵ filter, and the corresponding mass model was constrained from multi-band archival HST observations. The resulting H0 value is $$77.4\frac{{km}}{{s\,Mpc}}$$ with a precision of∼ 6 %.

We review the local determination of the Hubble constant, H0, focusing on recent measurements of a distance ladder constructed from geometry, Cepheid variables and Type Ia supernovae (SNe Ia). We explain in some detail the components of the ladder: (1) geometry from Milky Way parallaxes, masers in NGC 4258 and detached eclipsing binaries in the Large Magellanic Cloud; (2) measurements of Cepheids with the Hubble Space Telescope (HST) in these anchors and in the hosts of 42 SNe Ia; and (3) SNe Ia in the Hubble flow. Great attention to negating systematic uncertainties through the use of differential measurements is reviewed. A wide array of tests are discussed. The measurements provide a strong indication of a discrepancy between the local measure of H0 and its value predicted by Λ Cold Dark Matter theory, calibrated by the cosmic microwave background (Planck), a decade-long challenge known as the ‘Hubble Tension’. We present new measurements with the James Webb Space Telescope of >320 Cepheids on both rungs of the distance ladder, in a SN Ia host and the geometric calibrator NGC 4258, showing good agreement with the same as measured with HST. This provides strong evidence that systematic errors in HST Cepheid photometry do not play a significant role in the present Hubble Tension. Future measurements are expected to refine the local determination of the Hubble constant.

The jets image modelling of gravitationally lensed sources have been performed. Several basic models of the lens mass distribution were considered, in particular, a singular isothermal ellipsoid, an isothermal ellipsoid with the core, different multi-components models with the galactic disk, halo and bulge. The obtained jet images were compared as with each other as with results of observations. A significant dependence of the Hubble constant on the model parameters was revealed for B0218+357, when the circular structure was took into account.

In 1905, Henri Poincaré predicted the existence of gravitational waves and assumed their speed equal to the speed of light. If additionally the gravitational aberration would have the same magnitude as the aberration of light, we would observe several paradoxical phenomena. For instance, the orbit of two bodies would be unstable, since two attractive forces arise that are not in line and hence form a couple. This will be modelled by a nonautonomous system of ordinary differential equations with delay. In fact, any positive value of the gravitational aberration increases the angular momentum of such a system and this may contribute to the expansion of the universe. We found a remarkable coincidence between the Hubble constant and the increasing distance of the Moon from the Earth.

In 2000, Carlip showed that in general relativity gravitational aberration is almost cancelled out by velocity–dependent interactions. We show how the actual value of the gravitational aberration can be obtained by measurement of a single angle at a suitable time t* corresponding to the perihelion of an elliptic orbit. We also derive an a priori error estimate that expresses how accurately t* has to be determined to obtain the gravitational aberration to a prescribed tolerance.

Using the [O III] 5007 Planetary Nebula Luminosity Function (PNLF) distance indicator, we can double the number of known distances to nearby Type Ia SNe, and hence improve their zero point calibration while searching for systematic offsets in SN Ia luminosities between young populations (spiral galaxy) and old populations (ellipticals). We report three new PNLF distances and two new lower limits for five galaxies (NGC 524, 1316, 1380, 1448, & 4526) that have hosted well-observed Type Ia SNe. From the PNLF, we find a Hubble constant of 75 km/s/Mpc, whereas we derive H$_0$=72 from Cepheids and H$_0$=73 from SBF.