Research done during the previous century established our Standard Cosmological Model. Cosmology started to be considered a scientific discipline with the introduction in 1917 of the General Theory of the Relativity by Albert Einstein, which acts as theoretical framework for the development of the cosmological models. In the decade of 1920 Alexander Friedmann and Georges Lemaître suggested solutions to the equations of Einstein that provided dynamic universes, in expansion or in contraction. The discovery of the expansion of the universe carried out by Edwin Hubble in 1929 allowed for non-static models of universe that accounted for the observed expansion (the models of Friedmann-Lemaître that make use of the Robertson-Walker metric).
The idea that the universe might experience constant change locally and yet be on average invariable for very long times or eternity can probably be found among the ancient Greeks (Democritus). But the modern version dates firmly to 1948 and a pair of papers by Bondi & Gold (1948) and by Hoyle (1948), suggesting that the expansion of the universe, as implied by Hubble's and later work (Hubble 1929), is perfectly real but that additional matter is created at just the rate needed (about one atom per 106 cm3 per Hubble time) to keep the mean density constant, new galaxies being constantly formed from that new matter. A few of the observational objections to this picture would disappear if the new material consisted of about 1 helium atom for every 10 hydrogens, and perhaps 5 times as much in some form of dark matter capable of gravitational (and perhaps weak) interactions only, though DM was not, of course, part of the Bondi-Gold-Hoyle picture. Their intentions were at least partially philosophical, for instance to bring the process of creation within the observable universe.
The rate of creation required is very far below observability. But a pure steady state universe requires that the average mass and luminosities of galaxies, their clustering properties, and their propensity to emit strong radio fluxes must not change with redshift (time or place). The average age should be 1/3 of the Hubble time, making our Milky Way unusually old (though critics who had not thought through the issues tended to claim that the absence of young galaxies was the greater objection). But it was the requirement for a constant percent of galaxies to be strong radio sources that already cast serious doubt on the steady state model before 1960. Counting radio sources (Scheuer 1957, Ryle & Clarke 1961) was a disputed issue, but after the discovery of the redshifts for quasars (Schmidt 1963), there was no doubt that strong radio sources had been stronger and more common in the past (Schmidt 1968), providing evidence against the steady state model.
It was also not easy to reconcile the apparent brightness and angular diameters of distant galaxies with steady state requirements (since average properties must not change with time), and if creation was of pure hydrogen, then turning one-quarter of the material to helium in stars implied galaxies 5-10 times brighter than the ones we see.
In addition, clearly, there must not be anything found in the universe that could only have arisen under conditions very different from the present ones. Thus the 1965 discovery of the cosmic microwave radiation (Penzias & Wilson 1965) was the death of the steady state model for most astronomers who had not paid much attention to the earlier problems. There remains a small group of supporters of a short quasi-steady-state model, with much less of the simplicity enjoyed by the original one and the need still to doubt the cosmological nature of QSO redshifts as well as a need for intergalactic iron filings to thermalize the CMB and colorlessly absorb light from distant supernovae. We can admire their courage without having any desire to follow their ideas.
The discovery of the cosmic background radiation emitted by the hot gas when the universe was at 3000 degrees and had an age of about 380,000 years was a definite support for a general acceptance of a universe in expansion, with a finite age and an extremely dense and hot beginning, in what the physicist Fred Hoyle pejoratively called Big Bang. The name settled and the theory of the Hot Big Bang became the basic cosmological model, with some of its predictions ending up with clear observable successes, as the explanation (Alpher, Bethe & Gamow 1948; Peebles 1966) of the relative proportions observed in our local environment of the light elements (helium, deuterium and lithium).
The model itself was not exempt from some paradoxes, as the problem of the extreme homogeneity and isotropy among parts of the universe that had never been in causal contact (due to the finiteness of the light speed) or did not provide a convincing reason to justify that the density of matter and energy was so close to the critic value (flatness problem). With the introduction of the concept of inflation (Guth 1981) that suggests a phase of fast acceleration of the cosmic expansion in the early stages of the universe, some of these problems are solved, from the theoretical point, at the expense of introducing an additional hypothesis, which certainly is still not completely proved by observations.