For centuries astronomers had to content themselves with a 2-dimensional world with virtually no access to the depth of the Universe. The world unfolded before their eyes as though everything was taking place on the surface of a spherical envelope with few exceptions for the nearest sources, such as the Moon whose nearness was made obvious from its repeated passages before the Sun (solar eclipses), the planets or the stars (occultations). The size of this sphere was arbitrary and could not be gauged, let alone the idea that the stars could lie at different distances. Until the 17th century a reliable estimate of the true distance to the Sun and of the size of the Solar System remained out of reach, although a good scale model could be accurately devised and actually crafted in the form of delicately adorned orreries (but not all were on-scale).
Regarding the sidereal world and the immense vacuum lying beyond Saturn before reaching the first stars, some realistic ideas started emerging a good century later with the assumption that stars are Suns and share more or less the same luminosity. Gregory, Huygens among others came to numbers that at least hinted at the immensity of the world lying beyond the solar system. However the first indisputable stellar distances free of any physical assumption about the nature of the stars came out in 1840 through three independent labours, among which that of F.W. Bessel stands out. Once this first direct step has been mastered astronomers developed gradually a whole set of methods to ascertain the distances of celestial objects, each new step going farther in the cosmos and depending on the reliability of the previous rungs.
This short review aims at an audience of scientists with no particular astronomical background beyond the general knowledge shared by every physicist. Simple and basic formulas that would not appear in an astronomical research paper are given and explained. Only the principles of the methods are provided, illustrated on simple cases, leaving out the real difficulties which are the daily bread of practitioners. The book [1] by M. Rowan-Robinson provides a more technical and comprehensive review of the subject from stars to cosmological distances. Published before Hipparcos and HST, the content is a bit outdated but the description of the issues and the astronomical principles are still valuable and could be complemented with the more recent review of S. Webb [2]. At the solar system level the monograph [3] by A. van Helden is the best reference for the historical coverage from Aristarchus to Halley, but includes nothing relevant for the modern period.
The text is organised in two major sections. The first deals with distances within the solar system with the length of the astronomical unit in kilometres to its recent conceptual mutation to a defining constant with a fixed relation to the SI unit of length. The second part covers the scale of the Universe from the stars to the cosmological distances, with a particular emphasis on the first fundamental rung of the ladder completely rejuvenated over the last twenty years with the two ESA astrometry satellites Hipparcos and Gaia.