When is it required or permitted to sample below Nyquist rate
First of all, let's get rid of the Nyquist rate misconception. 

People are usually taught that the minimum sampling frequency needs to be twice the frequency of the highest frequency in the signal. This is completely false! 

What is true is that if you have a "full" spectrum, and by full, I mean that it completely uses up all frequencies between the lower edge of its bandwidth and the upper edge of its bandwidth, then you need to have the sampling frequency that is at least twice the bandwidth of the signal. 

  

So in the picture here, the sampling frequency needs to be at least 2*(Fh-Fl) in order to get the spectrum.

You also need to keep in mind that, after you do the sampling, all information about the actual frequency is lost in the sampled signal. This is where the entire story about the Nyquist frequency comes into play. If the sampling frequency is twice the highest frequency of the signal, then we can safely assume (as we're often trained to do subconsciously) that all the frequencies in the sampled signal are between zero and one half of the sampling frequency.
 In reality, the spectrum of the sampled signal is periodic around Fs/2 and we can use that periodicity in order to achieve lower sampling rates.
Take a look at the following picture:

The area between the 0 and Fs/2 is the so-called first Nyquist zone. This is the area where we're doing the "traditional" sampling. Next take a look at the area between the Fs/2 and Fs. This is the second Nyquist zone. If we have any signals in this area, their spectrum will be sampled and its spectrum will be flipped, that is to say, the high and the low frequencies will be inverted. Next, we have the third Nyquist zone, between the Fs and 3Fs/2. Signals here, when sampled, will look as if they came from the first zone and their spectrum will be normal. The same goes for all the other zones, with the rule being that the spectrum of odd-numbered zones is normal and the spectrum of even-numbered zones is inverted.   

Now this goes against the "traditional" rules about aliasing, since aliasing is usually taught as some evil monster coming to eat your signals away and that you have to use the low-pass anti-aliasing filters to get rid of it. In real life, this isn't how things really work. The anti-aliasing filters can't actually prevent aliasing, they just bring it down to the level where it doesn't matter any more.
 What we really want to do instead is to eliminate any strong signal from Nyquist zones that are not of interest and let through the signals from the Nyquist zone that is of interest to us. If we're in the first zone, then a low-pass filter is fine, but for all the other zones, we need a band-pas filter that will allow us to get the useful signals from that zone and remove junk that we don't need that's coming from the other zones.  

So let's take a look at this example:

Here we have a signal in the third Nyquist zone that is being let through by a band-pass filter. Our ADC will need to only have the sampling frequency of twice the bandwidth of the signal to reconstruct it, but we always need to keep in mind that this is actually a signal from the third zone, when we need to calculate the frequencies inside of our signal. This procedure is often called bandpass sampling or undersampling.  

Now, after all this exposition, to answer your question when:
Well, let's take a look at radio, perhaps something in the microwave spectrum, maybe WiFi. A typical old-style WiFi channel might have 20 MHz of bandwidth, but the carrier frequency would be around 2.4 GHz. So, if we take our naive approach to sampling the signal directly, we'd need a 5 GHz ADC to see our signal, even though we're only interested in particular 20 MHz of spectrum. A 5 GHz analog to digital converter is something that is very complicated and expensive and it requires very complicated and expensive design too. On the other hand, a 40 MHz ADC is something that is not as "magical" as a 5 GHz ADC.
One thing that needs to be kept in mind is that, although we could in theory capture out signal with a 40 MHz ADC, we'd need very sharp anti-aliasing filters, so in practice we don't really want to run the sampling frequency too close to the bandwidth. Another thing that's also overlooked is that the circuitry of a real-life ADC behaves as a filter on its own. The filtering effects of an ADC need to be taken into account when doing band-pass sampling. Quite often, there are special ADCs with bandwidths much wider than the sampling rate that are specifically designed with band-pass sampling in mind.

Finally, there's the other side of the story as well called compressed sensing. I'm not an expert in that, and it's something that's still a bit new, but the basic idea is that if certain assumptions are fulfilled (such as that the spectrum is sparse), we can sample at frequencies even lower than twice the bandwidth of the signal.