Straight up the Answer is almost certainly NO, since unless there is some way of restricting access to a site, any person can put on dive gear and go dive without any experience/training. I am not aware of any countries with laws prohibiting scuba diving in any form except for reward (commercial gain).
The more acceptable answer is, whenever you enter any overhead environment most divers/dive clubs/dive schools will require you to have appropriate training and experience. In the Light Zone, which is defined as linear 40m taking depth and penetration distance into account, the requirements are less strict and less enforced.
There are several organizations teaching Cave diving courses and they normally offer a few levels depending on how far you wish to penetrate and the time you will spend underwater. On a basic level you will have a Cavern course(lightzone), Basic Cave and Advanced Cave course. But there are many variations and add on courses, like using underwater propulsion vehicles, decompression diving, etc.
All over the world you will find interesting overhead diving that requires more advanced training, if this is something that interest you speak to people that knows the areas you are interested in and that knows cave diving.
Some dive boats/tours/centres will have their own requirements based on local laws/recommendations/experience, so as ever check before you go! (Aravona)
(Let me just add that there are people doing dives on their own without training and without building up their experience gradually, this is not something I recommend at all. Many divers have lost their lives doing dives past their limits and their knowledge/experience. Spend the time to find a good instructor and learn the theory, the skills/drills and do the dives gradually. In the end you will be safer and have more fun.)
Wikipedia has a good section on cave diving and lists a number of cave diving spots all over the world. WIKIPEDIA - CAVE DIVING