  We survey and develop the most elementary model of electron motion introduced
by R.Feynman. In this game, a checker moves on a checkerboard by simple rules,
and we count the turns. Feynman checkers are also known as a one-dimensional
quantum walk or an Ising model at imaginary temperature. We solve
mathematically a problem by R.Feynman from 1965, which was to prove that the
discrete model (for large time, small average velocity, and small lattice step)
is consistent with the continuum one. We study asymptotic properties of the
model (for small lattice step and large time) improving the results by
J.Narlikar from 1972 and by T.Sunada-T.Tate from 2012. For the first time we
observe and prove concentration of measure in the small-lattice-step limit. We
perform the second quantization of the model.
