Time-Delay Cosmography: Measuring the Hubble Constant and Other Cosmological Parameters with Strong Gravitational Lensing

Multiply lensed images of a same source experience a relative time delay in the arrival of photons due to the path length difference and the different gravitational potentials the photons travel through. This effect can be used to measure absolute distances and the Hubble constant ($H_{0}$) and is known as time-delay cosmography. The method is independent of the local distance ladder and early-universe physics and provides a precise and competitive measurement of $H_{0}$. With upcoming observatories, time-delay cosmography can provide a 1% precision measurement of $H_{0}$ and can decisively shed light on the current reported 'Hubble tension'. This manuscript details the general methodology developed over the past decades in time-delay cosmography, discusses recent advances and results, and, foremost, provides a foundation and outlook for the next decade in providing accurate and ever more precise measurements with increased sample size and improved observational techniques.