Beginning with the attempts of Heisenberg and Pauli in the 1920’s, the subject grewat an astonishing speed, culminating in the remarkable predictive successof Quantum Electrodynamics.Different attemptsto bring the subjectto a sound mathematical footing (comparable to that of Quantum Mechanics)– whetheranalytic, operator-algebraic or geometric – have tastedonly partial success. These talks will try to give a bird’s-eye view of the mathematical areas (ideas)spawned by these attempts, keeping in view the recent book of Folland on this subject.