Modern airliners like the Airbus A380 and Boeing 787 have bigger windows than older designs, like the Boeing 737 and Airbus A320.
What factors have led to this change of window size, opposite to the trend of almost every other thing in the interior of the plane becoming smaller in size (like the seats).
In the early years of the jet age, which followed closely on the introduction of pressurized hulls, there was much fear of stresses specifically in the area of the window openings due to the loss of two DeHavilland Comets. Subsequent Comet hulls had different shaped, rounded, smaller windows and that trend continued in other pressurized, typically jet-powered, aircraft.
Only with modern CAD systems and new materials knowledge could windows be made larger again. The newest aircraft designs reflect that progress.
Improvements in Engineering knowledge are an important factor.
Going from the time of the first 737 to the A380, there was a huge development in engineering techniques. Better material knowledge, more advanced modeling methods (such as more advanced Finite Element Methods, see this link for the history of FEM methods, thanks @Gürkan Çetin), and better knowledge of structural optimization have improved the situation. Especially with windows, where a lot is happening due to stress concentrations, this improved knowledge makes a difference.
This means that bigger windows might still lead to an increase in weight, but there will be a less severe weight penalty than the same windows using B737 age techniques.
It is the use of composite materials in the aircraft body structure that has enabled larger windows.
Metals are a crystalline structure that over time tend to fracture along crystal structure defects. Some key areas (e.g. turbine blades in the engine) are actually created by growing a single crystal to obtain higher failure points. This is not possible over a structure the size of the aircraft frame.
When an aircraft is at altitude, it is like a pressurised balloon. its skin is under tension, and when it on the ground, that tension is removed. This is like constantly flexing a metal paper clip. The cross sectional area of the aircraft skin that carries this tension is reduced by windows - bigger windows put bigger strain on the airframe structure which tends to concentrate at corners. making corners rounded helps to spread this stress.
The composite materials used in new planes are fibrous in structure. This greatly reduces the risks from the repeated flexing, allowing larger windows (and as the rust risk is reduced, cabin air is allowed to be more moist with also aids in flight comfort of passengers)
Let me answer with a car analogy: In most cases, the next car model is larger than the one before. Similarly, the aircraft manufacturers make the windows a little bigger with every new model, just to have another argument why the new model is better than all the competition. This costs a little weight, but seems to be worth it for the airlines.
Every next model is larger than the one before. This depends on the common trends of a particular period. Chevrolet Caprice, for example, is 18+ inches smaller than its predecessors. However, I agree that newer models are marketed as being better than the ones before, although there can be exceptions.
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As well as the advances in engineering, it is worth considering the comfort of the passengers. It is often thought that the more natural light in the cabin the better the experience for the passengers. More light increases the feeling of space, and can even reduce the fatigue experienced as a result of air travel. Whether or not there is much science in this I do not know.
Some Aircraft manufacturers use the window sizes as a big selling point. Gulfstream aircraft, for example, allow in more light than any comparable aircraft due to their oval shaped windows. In an attempt to compete Dassault are putting more windows on their newer Falcon 8x and Falcon 5x models.
Both companies have also introduced windows in the galleys in some models (Dassault famously with the 'sky light') as this is typically the darkest and narrowest part of the cabin. In the cabin configurations most typical in Europe, the galley is at the very front of the cabin and is the first thing you see when you enter the aircraft. I think it is safe to say that it is much nicer to walk into a bright, airy space than a dark, narrow corridor.
Gulfstream 650 window details:
Sixteen large Gulfstream panoramic windows, each 28 by 20.5 inches/71 by 52 centimeters, allow abundant sunlight into the cabin, even in the galley. Every window has been repositioned higher on the fuselage to maximize viewing comfort*
http://www.gulfstream.com/aircraft/gulfstream-g650
Falcon 5X window & 'skylight' details:
Increased natural light does more than improve the view. It elevates your mood and enlarges your perspective. The 28 large, expansive windows of the 5X provide unbeatable luminosity.
The Falcon 5X offers an unprecedented direct view of the sky overhead through its skylight ceiling window. This unique window on the sky transforms your perception of space while providing natural light from above.
http://www.dassaultfalcon.com/en/Aircraft/Models/5X/Pages/overview.aspx
One of the important things I'd like to add to the previous answers is that the old aircraft used to have smaller windows to accommodate for the rate of cabin depressurization in case of an emergency. The old airplanes had much smaller windows to control the cabin depressurization rate in case of some emergency (some window actually breaking to cause hull breach). But with advancement in the design methods and manufacturing process and with the introduction of the newer materials for airplane manufacturing, it has become possible to increase the size of the windows without increasing the risk factor.
In addition to this, creating an ambient atmosphere for the passengers inside the airplane without increasing the cost of the flight has always been a priority. So, while decreasing the seat spacing to accommodate more passengers is being done to reduce flight cost, the increase in the size of the windows does not have determinal effect on the travel cost, and hence is favored to create a less claustrophobic atmosphere inside the plane.
