For as long as I've been a fitness coach (over 20 years) the concept of training your Core has been around. Starting in the early 90's this industry buzz word exploded in popularity and fitness programs sprung up around the concept.
The theory being that basically modern life had become to easy, we were no longer hunter gatherers or even hard working farmers. The end result was that he deep muscles surrounding our spine have become weak and and therefore are the cause of a lot of modern day back ailments. Taken a bit further certified trainers offered the notion that if you improved your core strength your athletic performance would also improve.
To some degree this is partly true. Life has become easy and a lot of sedentary people do suffer needless back pain as a result. Yes athletic performance can be increased if your core muscles need to be trained and you spend time strengthening them. The problem is that solutions being offered as a fix have become campy and insufficient in addressing the problem.
However, I would like to propose that training your core has become such an industry buzz word that it has lost it's true meaning. A quick review of late night infomercials will get you dozens of useless crappy items with the buzz word "core" involved in it.
Most people who come through the door of a gym start off saying things like " I want to train my core" but if you spend a few minutes probing deeper into their needs and desires you find out the truth. Most people mistake the concept of training your core as meaning " I have fat around my midsection and want to fix this problem".
Well, if fat loss is truly the goal the answer is more likely to be found in your kitchen than at the gym . Google (Paleo diet) if you want to address fat loss.
If you are truly interested in developing a stronger "core" I suggest compound barbell movements as a better alternative. A properly performed back squat, deadlift or overhead squat will do far more to train these muscles than any amount of time flailing about on a bosu ball or swiss ball. When these movements are performed with moderate to heavy weights they do far more to strengthen this area of your body and have the added benefit of giving you real strength. They will also do far more to increase your athletic performance then spending endless hours with the "ab bean".
If you would like more info on strengthening your body (and core) or are interested on losing bodyfat contact your local CrossFit gym.