Q:

Why do glass windows still exist? ( Why haven't they been replaced by plastics?) 

Glass is fragile and impractical to transport, install and repair.  Even worse, glass kills and hurts people when it breaks.  Falling to the streets like guillotines during earthquakes and bomb raids.  During wars people put tape on their windows to prevent shattering.  When that meteor exploded over Chelyabinsk, people got hurt by standing inside of a window watching the sky when the shock wave hit them. 
There are perfectly transparent plastics, for example the PET material used to make coca cola bottles.  Why aren't windows made out of that instead of glass (fragile ceramics)?  It seems to be much cheaper, safer and more practical to handle.  Is there any advantage at all to make windows out of glass?  Is this a billion dollar business idea, and if so, why haven't anyone realized it yet? 

A:

There are two main reasons why glass is still preferred over say PMMA.  
The first is durability.  As long as it isn't broken, the glass in a window can easily last for hundreds of years in good condition.  In particular it is a lot more resistant to scratches than comparable plastics and isn't really subject to much in the way of environmental degradation.  Windows are very prone to getting scratched when they are washed as they accumulate small particles of grit on their surface which gets rubbed around the surface during cleaning.  Even with scratch resistant coatings no transparent plastics get anywhere near the hardness of glass.  
Most glasses are also much more resistant to environmental degradation from sunlight and various chemicals in the environment.  Even the most resistant plastics start to discolour and become brittle over time.  
The second factor is stiffness.  Glass has a much higher Young's Modulus than PMMA.  In bottles etc which are stiffened by their shape this doesn't matter much but, as windows tend to be large, flat, thin panels stiffness is a big issue, affecting their ability to be sealed into their frames and their optical properties.  So a plastic window would need to be substantially thicker than a glass one to have the same stiffness with consequences for optical quality and cost.  
There may also be issues with gas permeability in the context of double glazed windows.  
In addition many of the safety concern raised in the question are addressed by laminated and tempered glass.  Tempered glass is heat treated to control internal stresses, making it significantly stronger than float glass with the additional benefit that if it does break the entire plate fractures into small granules rather than sharp shards.  Laminated glass consists of alternating layer of glass and a polymer film, producing a composite sheet with very high strength and toughness, potentially to the point where it can be usefully bullet resistant.  
Another aspect of this is that side and rear windows in vehicles are often required to be made from tempered glass for safety reasons as they can be safely broken to allow access and extraction of passengers in an accident if doors are jammed or inaccessible.  

A:

Here is a plastic (most likely PMMA) window, in a boat, after only 37 years.  
 
In addition to the obvious scratches, the outer surface has developed a cloudiness : possibly from degradation due to UV light, and (towards the LH end) you can see a cubelike pattern of stress cracks, rather like a toughened glass windscreen after a pebble hit it. 
You really can't tell what you're looking at through it. 
In this application, a PMMA window is still the best solution, rather than the difficulty of making a glass window fit the curve of the hull.  But it shows the limitations compared to glass, which remains usable for centuries. 
As far as replacing glass windows, the cost of a specialist will be high for any type of repair in any technology. 
But absent custom curves, the raw material (glass) is cheap : about £1/sq foot, much cheaper than perspex or polycarbonate, and much easier and faster to cut. ( Watching a professional exploit the special fracture properties of glass is impressive.  Contrast with the difficulty of cutting perspex or polycarbonate!) 

A:

What are the requirements for window glass? 

Resistance to moisture
Resistance to UV radiation
Resistance to cleaning agents
Very high transmission in the visible spectrum
(There are many more, from an engineering point of view)

Most transparent plastics are not resistant enough.  Plexiglas is one example which fulfills the requirements enough to be used that way.  See this brochure from evonic where they guarantee you 30 years without noticeable yellowing. 
The biggest problem with PMMA for household use is its "softness" (more prone to scratches) and mediocre resistance to cleaning agents as compared to glass.  Now, there are many different flavours of PMMA, and you can coat a Plexiglas sheet to make it more durable. 
PMMA has many advantages over soda-lime glass:

Higher transmission in the visible spectrum
Lower density
Easier processing
...

It is at the moment probably just not cost effective enough to make PMMA as resilient as glass for use as window glass (for household use!,  PMMA is used in many different application as a better alternative to the classical inorganic glass).   This may change in the near future, as material science and the transparent plastics industry progresses.