  We prove that it is relatively consistent with the usual axioms of
mathematics that all automorphisms of the Calkin algebra are inner. Together
with a 2006 Phillips--Weaver construction of an outer automorphism using the
Continuum Hypothesis, this gives a complete solution to a 1977 problem of
Brown-Douglas-Fillmore. We also give a simpler and self-contained proof of the
Phillips--Weaver result.
