For example, it follows from Lagrange's theorem that every finite group is periodic and it has an exponent that divides its order.
Explicit examples of finitely generated infinite periodic groups were constructed by Golod,[1] based on joint work with Shafarevich (see Golod–Shafarevich theorem), and by Aleshin[2] and Grigorchuk[3] using automata.
The existence of infinite, finitely generated periodic groups as in the previous paragraph shows that the answer is "no" for an arbitrary exponent.
Though much more is known about which exponents can occur for infinite finitely generated groups there are still some for which the problem is open.
An interesting property of periodic groups is that the definition cannot be formalized in terms of first-order logic.