Many-worlds theory was proposed by Hugh Everett III (1930 - 1982) in 1957 in his PhD thesis. The theory was origniallly called theory of universal wave function; the name, many-worlds, was dubbed by B. De Witt (1911 - 2008) . According to this theory, the wave function of the universe is a superposition of many, many different components

         Ψ=φ₁+φ₂+...+φ_n+...,

where each component φ_n represents a different parallel world. Each of these worlds evolves indepentently and do not influence each other; occasionally, they interfere.

The many-worlds theory is now clearly the best intepretation of quantum mechanics. Unfortunately, it is a minority view among physicists. I think that there are two main reasons: (1) there is psychological barrier for many to accept that other parallel worlds or realities exist; (2) many have not read Everett's long PhD thesis, where this theory is elaborated with rigor and details.  

Here is the original theis. However, this thesis is not an easy read as it is over 130 pages and Everett used many unfamiliar mathematical notations. So, I made an abridged version.