Since Bessel beams were introduced some twenty years ago, they have intrigued
the scientific community due to some of their apparently odd or strange
features. Some of them are even against the physical intuition. As examples of
this are: the fact of being non-diffracting, the central spot of the zero-order
Bessel beam could be smaller than the wavelength and that they are self-healing, i.e. they auto-reconstruct after being partially blocked by an opaque obstacle. Nowadays four families of these beams which are closed solutions of the Helmholtz equation are known. Besides plane waves and Bessel beams, other two possible families of non-diffracting beams have been predicted and demonstrated experimentally. Recently, another kind of non-diffracting beam of the (1+1)paraxial wave equation has been introduced namely, the Airy beams. Gradually,non-diffracting beams have become a matter of great interest due to their invariant propagating features and the wide variety of applications ranging from atom trapping to optical communications. In this work, The paper present the fundamental theory behind these beams showing that they can be as friendly and physically consistent as the very familiar plane waves. The paper, also reviews and discuss the main results and applications reported to date, revealing the mysteries of Bessel and other families of non-diffracting beams. Since Bessel beams were introduced some twenty years ago, they have intrigued the scientific community due to some of their apparently odd or strange features; some of them are even against the physical intuition.