Barefoot training is still popular, especially with footwear like Vibrams.
These minimalist forms of footwear allow you train “barefoot” in public gyms. Before you try this type of training, there are some things you should know.
First, you have to work up to being barefoot. If you wear shoes all the time, then you can probably start with lifting weights in these types of shoes. Then work up to jumping rope, which will build up your ability to tolerate short amounts of running.
The main reasons to go barefoot are to build up the little muscles in your feet and to correct poor mechanics. Poor mechanics get corrected by getting more feedback and avoiding pain.
If you heel strike when you run in shoes, you will not be able to in minimalist shoes because it will hurt. It may correct your form, but if you have a major foot dysfunction like flat feet you might want to check with someone who has knowledge of foot mechanics.
If you wear corrective devices like inserts, then it might take you longer to ease into bare foot training, but in the end it will help your foot muscles even if you do not correct all of your faults. Some people run all the time in these types of footwear, but they will most likely be a supplement to your training.