What we know about the universe is that it is vast, old, and growing in size. How old? One estimate is that it is about 14 billion light-years old (http://www.space.com/25325-fermi-paradox.html). Take note; it is in "light years". So, it's definitely way more than 14 billion Earth-years old given the speed of light, which is ridiculously fast.
So, the universe is very very old. We have this postulate that there is life out there other than ours, and that way older civilizations exist. These civilizations might be more advanced, and therefore, they might have developed very powerful space technology, probably enough to have reached the Solar System and the Earth. Seems likely, right? Along this line of thought, Enrico Fermi, an Italian physicist, was said to have stated assumptions informally (http://www.seti.org/seti-institute/project/details/fermi-paradox), which I have noted down into these statements:
a. a civilization with modest rocket technology but with enough resources could rapidly conquer an entire galaxy
b. an empire could have conquered star systems within ten million years, enough to rapidly colonize an entire galaxy
Given these assumptions, Fermi thought that civilizations way older than ours, given the age of a galaxy and the universe, had more than enough time to make their presence manifested, say within the Milky Way galaxy. So, as Fermi put it, "where is everybody?" This is the so-called "Fermi paradox". Many solutions have been proposed to address this problem, and I think they were interesting, or rather, entertaining. However, at the bottom-line, what are the odds that Fermi's assumptions are true. This leads to the Drake Equation (http://www.seti.org/seti-institute/project/details/fermi-paradox):
N = R* • fp • ne • fl • fi • fc • L, where
N = the number of civilizations in the Milky Way galaxy whose electromagnetic emissions are detectable;
R* = the rate of formation of stars suitable for the development of intelligent life;
fp = the fraction of those stars with planetary systems;
ne = the number of planets, per solar system, with an environment suitable for life;
fl = the fraction of suitable planets on which life actually appears;
fi = the fraction of life bearing planets on which intelligent life emerges;
fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space; and
L = The length of time such civilizations release detectable signals into space.
The estimates for the value of N are not reliable as of this moment given that we only have assumptions so far, and we have no strong empirical evidence that life exists in other planetary systems. Life is indeed very challenging under this research area right now, and while we are still groping for empirical evidence, it seems fun to think about the solutions to the Fermi paradox. But wait, how feasible the assumptions are?
One may think that it is not possible for a single alien to have a life span of a million years. Then, there might not be enough fuel for an unmanned spacecraft to travel a massive distance, and perhaps by traveling for more than a million years, the spacecraft may have disintegrated. The idea seems absurd given the current technology. However, we are talking about millions of years of advanced technology, about the "what if's". Perhaps, the obstacles posed by the theory of relativity have been conquered. However, in the case of Fermi's reasoning, the spacecraft involved can be slow; there's no need to develop a spacecraft that is as fast as the speed of light. If you are familiar with the Bracewell-von Neumann probes, a civilization should just develop these probes. They need not be fast, but they should be autonomous and self-replicating. In order to replicate, they need to find and retrieve the necessary materials from space, e.g. asteroids. The number of probes can grow exponentially, and a galaxy can be explored in "just" less than 10 millions of years. Compared to the age of a galaxy, which can be billions of years old, the exploration can be considered quick. So, we are left with the question whether the Bracewell-von Neumann probes are easy to build.
In tackling a solution to the paradox, we should assume that building the probes is easy or possible. The possible solutions, therefore fall into three categories (http://abyss.uoregon.edu/~js/cosmo/lectures/lec28.html): (1) "they are here", (2) "they exist but have not yet communicated", and (3) "they do not exist". I have encountered many interesting solutions, and you might want to check a less boring take on the subject matter here: http://waitbutwhy.com/2014/05/fermi-paradox.html. Among these solutions, what I consider to be one of the mind-boggling ones is that all of us have been subjected to a superbly sophisticated simulation by the aliens, which makes the state of our existence similar to that in the movie "The Matrix".
