What challenges needed to be overcome to create (blue) LEDs? In light of today's announcement of the 2014 Nobel laureates, and because of a discussion among colleagues about the physical significance of these devices, let me ask:
What is the physical significance of blue LEDs, which challenges had to be overcome to create them? Why are materials with the band gap necessary for blue light apparently so rare/difficult to manufacture?
I know it took decades to create blue LED after Holonyak discovered the first red ones, so there must have been some obstacles, which were maybe also important for other areas of research - otherwise I wouldn't understand why the inventors of the blue LED got a prize that the inventor of the first LED didn't. 
Wikipedia has something to say on the topic:

Its development built on critical developments in GaN nucleation on sapphire substrates and the demonstration of p-type doping of GaN.

However, I'm asking myself why this is "critical" and why this was difficult.
 A: The book The Blue Laser Diode: The Complete Story deals with the issues of p-type doping of GaN. 

The difficulty of growing high quality GaN crystalline films lies in the problem of finding a suitable substrate material. (...)

The link above points to the chapter you may be interested in.
A: The "critical" part was in finding and producing a structure with a large enough bandgap to produce blue photons.  The first LEDs produced relatively longwave infrared (IR) photons, which have far less energy than the green or blue photons now available from LEDs.  In general, the larger the desired bandgap, the  harder it is to manufacture a suitable material.
That said,  Claudix' reference is a good book to check out.
A: The Nobel website scientific background is good. Basically, when you try to make gallium nitride, you usually end up with a material that is (1) chock-full of defects, and (2) n-doped (even when you were trying to p-dope it).
So blue LEDs required

*

*The invention of MOCVD technology for growing crystals (early 1970s);

*Finding the right recipe to grow good GaN by MOCVD (i.e., use a sapphire substrate, start with a low temperature step then switch to high temperature, etc.) (mid-1980s);

*Finding the right recipe to grow p-type GaN (what dopant to use (Mg), in what concentration, and what annealing / treating recipe to use to make the Mg dopants actually work and reduce the number of unintended n-type dopants that were canceling it out) (early 1990s);

*Once all that was in place, find good structures to make LEDs (e.g. if you can also grow InGaN then you can make quantum wells) (early-to-mid 1990s).

All these steps required not only painstaking trial-and-error but also lots of insightful analysis and careful measurements to diagnose the problems and discover how to fix them. :-D
Sidenote: I think it's really cool and exciting that this line of materials-science research is not finished yet. As you alloy more and more indium into indium-gallium-nitride, the defects get even worse and p-doping becomes even harder. There are now lots of people working on overcoming these problems. Each year it seems that someone comes up with a materials-processing breakthrough that allows them to use a few more percentage points of indium.
With enough indium, the bandgap would shift from blue to green (with MUCH more indium, it shifts all the way to infrared). So this research could potentially lead to a much more efficient green LED, and even better, the long-awaited green diode laser, which would have myriad applications e.g. in display technology. (You've seen green laser pointers, but these are complicated devices that use infrared lasers and nonlinear optics. A green diode laser, if it existed, would be cheaper, more rugged, smaller, and way more energy efficient.) Also, if you could use more indium, InGaN-GaN becomes a promising candidate material system for tandem solar cells.
A: I wanted to post this as a comment, but it grew too long.
The final question that was a bit hidden, but that several other users seemed to be also interested in, would be about why the blue LED was maybe so much harder to construct than the red one. Reading through Steve B's link to the nobel scientific background provides me with enough information that I can try to answer the question for anyone else, who is interested. I'm getting all my information from there: 
Apparently, looking at np-junctions it was clear to many people that they could be useful to create light sources. The first infrared lasers/LED were then created at almost the same time when the first red LED was announced. The only difference is the band-gap, which was bigger in the case of the LED, because Holonyak used a clever combination of the existing approaches ($\mathrm{GaP}_x\mathrm{As}_{1-x}$ instead of $\mathrm{GaP}$ or $\mathrm{GaAs}$). The LED were not very efficient at first, what was making them efficient were the techniques to use heterostructures and quantum wells - the first was awarded a nobel prize in 2000. 
In this sense, making a red LED was definitely a big breakthrough, but it was also in continuation of the current research at the time. 
The blue LEDs, however, are different: Some people had proposed to use $\mathrm{GaN}$ as a basis already in the 70s, but nobody could get it to work. Also, additional challenges had to be overcome listed as points 2.-4. on Steve B's list, which were essentially all performed by the three people who were awarded this year's Nobel prize.
So maybe it's more like this: If the invention of the first LED merits a Nobel prize (whatever that means), then certainly so would the invention of blue LEDs.
A: Additionally, blue was the last of the primary colours so its invention made the production of white LEDs possible. Ordinary lamps could then be replaced with extremely energy efficient LED alternatives.
