  We prove finite time blowup of the Burgers-Hilbert equation. We construct
smooth initial data with finite $H^5$-norm such that the $L^\infty$-norm of the
spacial derivative of the solution blows up. The blowup is an asymptotic
self-similar shock at one single point with an explicitly computable blowup
profile. The blowup profile is a cusp with H\"older $1/3$ continuity. The
blowup time and location are described in terms of explicit ODEs. Our proof
uses a transformation to modulated self-similar variables, the quantitative
properties of the stable self-similar solution to the inviscid Burgers
equation, an $L^2$-estimate in self-similar variables, and pointwise estimates
for Hilbert transform and for transport equations.
