Fermi Paradox was first pointed out in 1950 by the physicists Enrico Fermi and Michael Hart based on probabilistic arguments about the existence of alien civilizations, such as the Drake Equation: if there really is a great amount of technologically advanced extraterrestrial civilizations, as estimates seem to predict, why until now haven't we seen any direct evidence of the existence of even one of them? This becomes even more problematic if we consider that several star systems, and even an entire galaxy, could be colonized in a few million years, a very short time that should have already passed: if even one advanced civilization ever existed before us in the Milky Way, we should expect the entire galaxy to belong to it by now.
There are no other civilizations
It's entirely possible, of course, that probabilistical arguments in favour of intelligence in the cosmos are simply wrong. There is a great uncertainty of the actual probabilities of each passage: planets suitable for life might be much less common than we hope (see the Rare Earth Hypothesis); even if those planets are common, the origin of life might be unlikely; even if it's not, life might be very susceptible to mass extinctions, and disappear easily or fail to develop intelligence; even if intelligence is common in the Universe, intelligent species might fail to develop highly technological civilizations, dying out earlier or existing in a relatively primitive way (after all, even today several cultures on Earth have maintained their traditional technology since Paleolithic).
According to a somewhat more somber argument, technological civilizations might in fact be common, but they could have a tendency to destroy themselves either indirectly, through pollution and resource depletion, or directly, through war or accidental technological disasters (e.g. the grey goo scenario). This view was held by several scientists, such as Carl Sagan, during the Cold War; some suspect that any intelligent civilization is bound to be on some measure aggressive and xenophobic, as these traits would be naturally produced by evolution. Besides, intelligent species tend to occupy a higher trophic level in their ecosystem.
We cannot communicate
While the search for extraterrestrial civilizations is mostly based on radio waves, there are already other forms of communication that would not be intercepted so easily: for example, directional narrow-beam microwave and lasers, that we couldn't receive unless they specifically passed through Earth. Even within radio waves, SETI and similar projects only consider specific frequency ranges.
Then there's the issue with understanding the messages. Their comprehension could only be based on redundancy, that is, the repetition of elements within the message; a perfectly compressed or encrypted message, that contained only truly essential information, would be completely unrecognizable from random noise to any observer lacking the compression algorithm.
Physiology and psychology add yet another layer of difficulty. Visual messages such as the graphic parts of the Arecibo Message or the Pioneer plaques might be unintelligible to aliens with a primary sense different from sight, such as echolocation, smell or electroception. Math and physics will probably be the only common ground understandable by both, though they could be expressed in a very different form; something like Lincos might be the language of choice for interstellar communication.
Civilizations are too far away
Even if extraterrestrial civilizations really are common, the sheer size of the Universe could be enough to deny a meaningful communication: even with the optimistic values, given here, of 14 highly technological civilizations per year (N*) in the Milky Way, each lasting one million years, they'd still be, as an average, 330 light years apart from each other. Barring faster-than-light travel, such a distance would be prevent any contact besides radio waves and similar forms of communication, and even those would require a wait of nearly seven centuries between sending a message and receiving its response. Interstellar travel and colonization is not likely to make far away civilizations significantly closer to us.
Anyway, the fact that there are ETI relatively close to us does not necessarily guarantee that we'd know. Earth-like television and radio broadcasts become unreadable for observers such as rthe Arecibo telescope after only 0.3 light years, and programs such as SETI have searched only for a very small portion of the near space thus far.
Besides space, we also need to consider time. Even if every star system near to us developed a civilization in some point of its history, the probability of those civilization overlapping with ours are very poor: Homo sapiens occupies only the 0.004% of the past history of the Solar System, human civilizations the 0.0001%, and modern industrial civilization the 0.000004%. Active search for ETI is even younger than that. Of course, we don't know how much civilizations can last: perhaps they endure until the death of their star, perhaps even beyond that, perhaps they die out before (see above). Even if the other civilizations are ancient, ours is still young.
A civilization, though, doesn't need to die out to become unapproachable (see below). What if they simply stop emitting signals? After all, most of mankind "footprint" in space has been produced by leaked radio waves: since they represent a loss of energy that never reaches the intended target, economy tends to reduce them (less wasteful optical fibers and laser transmissions have already begun to phase out radio waves on Earth). As humorously explored here, the range of time in which signals are sent in space might be very short even compared to the civilization's lifespan.
They don't want to interact
Another possibility to consider is that near civilizations, while able to communicate with us, simply don't want to. Reasons for such a choice would depend from the ETI's psychology and culture, and might appear incomprehensible to us: interstellar contact could be prohibited by political, religious, philosophical or economical factors. One example: benevolent ETI with a great technological advantage over us might perceive any possible relationship with us to be too unfairly one-sided to be acceptable. As an alternative, they might just be too stranded in debates about what kind of signal send to other civilizations, or whether send one at all, to establish a meaningful communication.
Other possibilities: they could be too much immersed in virtual reality to care about the rest of the Universe; they might be afraid of meeting hostile entities; they might hold a religious belief that they're the only intelligence in existence; they might be observing, themselves unobserved, the development of Earth without interfering with it (zoo hypothesis).
Filters: some evolution steps are hard to take
One evolution step (nothing to organic matter, organic matter to bacteria, bacteria to intelligent life, intelligent life to galaxy trip...) could be harder to take than we think. If we already stepped towards, we are lucky but we are the only one. If the step is in front of us, perhaps we will never meet any other civilizations.
Life was not possible one and a half billion year ago
Although the universe is stable for a while, it constantly changes. The cosmic microwave background constantly decreases. The composition of the stars is different in relation to their age. A physical phenomenon (radiation, etc...) could be so intense in the past that it would have destroyed any organic matter. Then its intensity has gradually decreased and one and a half billion year ago life started to be possible. This can explain why no civilization is older enough to visit us and why we haven't appeared on a older planetary system.