From the beginning of the 1950's, Werner Heisenberg was occupied with what one could call "the new problem of elementary particles". This problem arose when large amounts of unknown and short-lived particles were found in the cosmic radiation and in high-energy accelerators. Eventually, as a result of the discoveries and inventions of many scientists, several of which became Nobel Laureates, we have today what is called the Standard Model of particles and forces. This theory explains almost everything about the plethora of particles that are produced by the cosmic radiation and in large accelerators. But the road to this goal has left quite a number of fascinating ideas behind, including some ideas of two of the most frequent early Lindau participants, Werner Heisenberg and Paul Dirac. As young men, these two physicists not only shared the night train from Copenhagen to Stockholm in 1933, arriving (un-married and instead accompanied by their mothers) on December 9 to pick up their respective Nobel Prizes. They also refused to miss any Lindau physics meeting as long as their health allowed. Dirac (1902-84) managed to lecture at 10 meetings, the first one in 1953 and the last one in 1982. Heisenberg (1901-76) lectured at 7 meetings in physics and 1 in chemistry, his first lecture in 1953 and his last in 1971. What also bound them together was the choice of subjects for their lectures. Both of them freely shared their latest bright ideas with the young researchers and students at the meetings. Some of these ideas have survived and some have been left behind. The latter category includes, e.g., Dirac's so-called Large Number Hypothesis described in his lecture of 1979. It also includes, e.g., Heisenberg's 1959 lecture on a unified theory of elementary particles, the present one. This does not mean that the lectures are less interesting. On the contrary, they act as time machines and give us fascinating glimpses of the research frontiers at the time of the respective meetings! In his introduction, Heisenberg pays respect to a colleague theoretical physicist and Nobel Laureate, the then newly deceased Wolfgang Pauli (1900-1958). Pauli, whose personality was very different from both Heisenberg's and Dirac's, only participated in the1956 Lindau meeting and gave no lecture. But for some time before he passed away, Pauli collaborated with Heisenberg on the theory that is described in the present lecture. Heisenberg delivers his lecture in a very energetic way and it is received by the audience with a very energetic applause. It is quite interesting to note that the quality of the tape recording is so good that we are even allowed to hear the crayon that Heisenberg is using to write his mathematical expressions on a black (or green?) table!