With so many things being ruled out to travel with, or to be stored in the correct manner anything with a battery or regarded as electronic will more than likely have rules attached for air travel.
It will come as no surprise that traveling and smoking is now completely banned, and it is therefore the same with air travel and there are some airlines that regard it as illegal.
For a short period of time, you could use e-cigarettes that the airline would provide you with when vaping first started but this is now no longer the case and there are a variety of reasons for this that are completely understandable.
Being conscious of your fellow air travelers is one of the main reasons and the dangers that carrying a working battery will mean to the aircraft and controls within it. We believe that phones, internet, and many other waves could affect the aeroplanes controls and thus it is safer for everyone to only carry on themselves what is deemed appropriate.
We do however know that you can take your vape with you as long as it is a part of your hand or cargo luggage so, you can continue to use it once you have landed and gone through the security and customs exercises. Each country may be different and therefore it may be checking with your airline before you travel as they will advise you on their own policies.