The commonly known Yang-Mills theory should be more precisely called as Yang-Mills-Shaw theory. The reason is as follows:

3.4. Yang-Mills-Shaw theory-- www.hull.ac.uk/php/masrs/reminiscences.html

A footnote on p.37 of Part II of my (Ron Shaw, University of Hull) dissertation reads:

"The work described in this chapter (ch.III) was completed, except for its extension in Section 3, in January 1954, but was not published. In October 1954, Yang and Mills adopted independently the same postulate and derived similar consequences."

But although their publication date was in 1954, Yang and Mills must have priority since it seems that their research was completed in 1953. 

The idea for chapter III of part II came to me in a flash while reading a manuscript of Schwinger's, which I found left lying around in the Philosophical Library in Cambridge. In it Schwinger showed how invariance of the Lagrangian under general gauge transformations required the introduction of the electromagnetic field. This of course was not new (though possibly it was to me at that time), but Schwinger's manuscript used real spinors, and so the usual U(1) invariance appeared instead as SO(2) invariance. Being familiar with Kemmer's work on invariance under "special" isotopic spin transformations it seemed to cry out to see what would happen if I changed Schwinger's (abelian) SO(2) to the (non-abelian) isospin SU(2).

I showed my generalization to Salam in early 1954, but in a rather disparaging way, since I did not doubt at that stage that the new non-abelian gauge fields would require particles to have zero mass, and such particle did not appear to exist in nature. Later on in 1954, Salam showed me the paper by Yang and Mills. Salam still wanted me to publish my contribution, but I never did. On many occasions (the 1962 Istanbul Summer School on Group Theory in Physics, the Schrodinger Centenary Conference at Imperial College in 1987, ... ) he publicised my independent discovery. In his Nobel Prize Lecture 1979, reprinted in Rev. Modern Phys. 52 (1980), 525-538, there are (see below) several references to Yang-Mills-Shaw theory. I have also recently come across a letter from Salam to me dated 1 Oct 1988 (in connection with the submission of a paper of mine to Proc. Roy. Soc.) in which he again refers to Yang-Mills-Shaw theory, and reminds me:

"I still remember asking you to publish this and you were very shy at that moment because you thought Yang-Mills had published it already although you had done the work independently."

However most physicists just refer to Yang-Mills theory --- and actually I am quite glad of this! I like a quiet life, and would not have enjoyed being pestered throughout the decades by lots of queries from researchers expecting me to be up to date with latest developments.