Why is amplitude-modulated radio broadcast (AM) still active?

In a nutshell: One antenna will give you a usable radius of 100...1000 km, depending on the power used.

In Germany, for the example of my favored news station Deutschlandfunk, we used to have two long-wave AM stations (153 and 207 kHz, IIRC), and I do miss them every once in a while. The one at 207 kHz covered pretty much all of Southern Germany, and while I admit that the quality was low (as in: landline telephone-ish low), you could listen to the program with no trouble, anywhere in your house, and understand every word well.

Now, for terrestrial distribution, they just use FM, which works in a few small places only, or you could try DAB+, and I'm not sure if the latter works in all places. I do miss the robustness and the beautiful simplicity of long or medium wave AM.

It's not so much the type of modulation (AM vs. FM). It's the low-ish frequencies that tend to work well over wide areas and even through big walls, for example if you're downstairs.

It's not true that no one ever listened, and in contrast to North America, for example, Germany used to have only very few good stations on AM in the decade before they pulled the plug on it, which gives you another very important reason why few people listened.

A personal note: It twists my stomach to see how AM has already vanished, and to know that some want to abandon analog FM as well.

If you were to get cynical, you could argue there is some strong political will to seriously srew up anything terrestrial for good, at least in Germany. A bit off-topic here, and a rant, but terrestrial TV broadcasting shows you how bad it can become, and it's a fine example of unclever engineering: Analog terrestrial TV was shut down not long ago, in the early 2000s, with DVB-T as a replacement. Soon, (mostly private) stations stopped broadcasting on DVB-T, and now, DVB-T2 is about to be introduced, and of course, it's not backwards compatible to DVB-T, so any DVB-T receiver will be a piece of useless junk^TM very soon. Considering the beauty of analog TV, this is sickening. There was black-and-white TV. Then they figured out how to put color into the signal while black-and-white receivers would still decode black-and-white and the new color TV signal, and color TV receivers would decode old black-and-white signals just as well as new color TV signals. Then, they put all other sorts of fancy stuff into the signel (stereo, videotext, ...) and everything was still forward and backward compatible. That's what I call good engineering, even more so if you put it into the context of its time and consider how advanced things were with respect to what was possible with the available technology.

Besides that also in Germany you can very well listen to AM stations, although the transmissions may not be as crisp clear as wide FM stations. There are several reasons that come to my mind why they might still operate:

• Old AM equipment may just still be there and function, and investing in replacing with FM is prohibitively expensive
• An FM frequency slot may not be available or the license be too expensive (FM is done usually on quite different frequencies)
• The range where you can receive (especially at night) can be many times as big as for FM, making reaching the same audience difficult to expensive to impossible (depending on the range)
• For political reasons you want to be received in other countries, but can not output that much FM power
• You want to bring radio of your language to a far away different country, but can't afford the usually many times higher prices FM license
• Sometimes an AM station is just transmitting the same "data" as some FM station, just for reachability or similar reasons

Others have mentioned that old equipment is often still used. Just to expand a little bit upon this: Unlike many of new technologies of today, where the lifetime of the technology itself is very short, AM radio comes from an era of long-lasting technologies. Much of the infrastructure is quite old and is still working fine.

It should be noted that this goes both ways: For broadcasters as well as for the audience. Pretty much every radio made in the last 80 or so years will be capable of receiving at least medium-wave AM stations. It's been only in the last 10 or so years that VHF-only radios have gained popularity.

The result of that is that AM is a well-entrenched technology which is difficult to update.
At first the issue was technical: good sounding FM transmissions need much more bandwidth than AM ones do, which, at frequencies where one might encounter AM radio, is simply not available.
Afterwards, with technology progress, new modulation types became available which resulted in standards such as Digital Radio Mondiale. In a standard 9-kHz European long and medium wave channel, it could provide much better sounding audio than AM, send additional data, similar to what's available on DAB/DAB+ or have two AM-quality voice programs at the same time. The downside of this was that, like for DAB, you'd need a new receiver.
This is where the problem appears: Because there are so few receivers available, stations don't want to upgrade their equipment to new standards, and on the other hand, because there are so few broadcasters, manufacturers don't want to start making DRM receivers in large amount. So today, even if you wanted to buy a DRM receiver, it will be difficult. Of course, there's the price as well, because the DRM receivers are much more expensive than usual AM/FM receivers and even FM/DAB receivers.

On the other hand, I suspect that this expectation of technology change has also reduced acceptance of backwards-compatible upgrades to AM broadcasting. Only a few stations support the AM Signalling System, which allows transmission of station identification and other information, similar to RDS on FM. I've also not seen any receivers that can decode AMSS.

Furthermore, medium-wave AM is used for regional and intra-continental coverage, while short-wave is usually used for intercontinental coverage. Countries that need medium-wave to cover their territory usually have relatively low population density, are poor or are a combination of both. The result is that the countries that need this technological update the most, can't afford it and need to stick with old-style AM.