The search for extraterrestrial life in the Universe has been something that has occupied the collective mind of humanity for generations. Since the 1600s, humankind has realized that the Universe is far bigger than was originally thought. With the existence of other planets, the possibility of extraterrestrial life arises. So it is natural to ask 'How likely is it that alien life exists?'.
Definitive proof for the existence of alien life has yet to be found. However, Astronomers have made attempts to at least create a framework for answering this question. The question of how likely life is to arise in our own galaxy was first framed by Frank Drake, in what has come to be known as the "Drake Equation". This equation governs how likely life is to arise and be a communicating civilization by considering a set of likelihoods, and then combining them.
The Number of Stars in our Galaxy – Drake began with all possible stars in our galaxy. Recently this number has undergone revision, and it is possible (although being investigated still) that there are more than one trillion stars in the Milky Way (as opposed to the previous estimate of 400 billion).
The Fraction of Stars with Planets – Drake then narrowed down the possibilities by saying that the only star systems which could have life would be ones which had planets orbiting around the star. The search for extrasolar planets has made great progress in finding planets orbiting around Suns other than our own. To date they have discovered more than 400 of these "exoplanets".
The Average Number of Planets in the "Habitable Zone" – The Habitable Zone is the set of possible distances from a star where the temperature is right to have liquid water. Here on Earth, it seems that there are so many reactions in organic chemistry which require water as a substance that it is difficult to conceive of a situation where life could arise without liquid water. Therefore, Drake chose to make this a limiting factor in his equation.
The Fraction of Planets on which life arises – This number is, obviously, not very well known yet. Biology is still catching up to the point where it can sensibly speak about the probability of a particular type of life form arising. As fields like Biology and Biophysics work to quantify the probability of life coming out of a particular mixture of chemicals, energy, and time, we will be able to more accurately "pin down" this value.
The Fraction of Planets with life that has an evolution to intelligent life. A planet may develop life, such as a species of bacteria, which is considered alive but still not ever capable of communication with other civilizations. Since Drake's equation is supposed to govern the likelihood of intelligent and communicating civilizations arising, it automatically eliminates those planets which have life that is of too simple a form to communicate with other civilizations. Like the term before this one, this probability isn't known. The answer to the question "How likely is a species to become intelligent over time?" is one that is a matter for quantitative Evolutionary Biology. This field is still in its infancy, in terms of the ability to quantify results. However, advances in fields such as Computer Science and Mathematics lend themselves to helping in the quantification process. In coming decades, we should have the capability to determine this number much more precisely, thereby dropping another piece in the puzzle.
Finally, the last term is the fraction of a star's life for which it is communicative. If a civilization lasts, on average, for a very short period of time, it is not likely to be able to communicate with other civilizations before it is extinguished. If, however, it lasts for a very long period of time, the likelihood of contact with other civilizations is quite high.
The progress of Astronomy has allowed for a few of the terms in the Drake equation (Number of stars in our galaxy, Fraction of stars with planets, Average number of planets in the habitable zone) to be investigated. While these terms are still not known with a high degree of certainty, early attempts have started to bear fruit.
There is still much to be done, however, before the latter terms in Drake's Equation become amenable to analysis. Our lack of knowledge in the fields of Evolution and Biology currently give us no way to solve for some of the relevant terms. However, given the inevitable march of Science, it is quite possible that at some time in the not too distant future, we will have actual values for all terms in the Drake Equation, and therefore say with certainty just how many communicative civilizations there are in our galaxy.