The Backward Fokker-Planck Equation (BFPE) provides an elegant technique for solving a number of first-passage problems. After an introduction to the method, the usefulness of the BFPE will be illustrated by applying it
to a number of problems  of increasing complexity, namely
(i) the simple random walk with an absorbing boundary at the origin;
(ii) the simple random walk with an absorbing boundary moving with constant velocity;
(iii) the simple random walk with two absorbing boundaries moving at different velocities, and
(iv) the random walk in the presence of various types of "shear flow" and an absorbing boundary at the origin.
The last of these is a problem with very rich behavior, depending on the nature of the "shear flow".