  We develop a coordinate space renormalization of massless Quantum
Electrodynamics using the powerful method of differential renormalization. Bare
one-loop amplitudes are finite at non-coincident external points, but do not
accept a Fourier transform into momentum space. The method provides a
systematic procedure to obtain one-loop renormalized amplitudes with finite
Fourier transforms in strictly four dimensions without the appearance of
integrals or the use of a regulator. Higher loops are solved similarly by
renormalizing from the inner singularities outwards to the global one. We
compute all 1- and 2-loop 1PI diagrams, run renormalization group equations on
them and check Ward identities. The method furthermore allows us to discern a
particular pattern of renormalization under which certain amplitudes are seen
not to contain higher-loop leading logarithms. We finally present the
computation of the chiral triangle showing that differential renormalization
emerges as a natural scheme to tackle $\gamma_5$ problems.
