  We provide a review of some symmetry-related literature on the eikonal
equations $u_\mu u_\mu =0$,$u_\mu u_\mu =1$, where lower indices at dependent
variables designate derivatives, $\mu=0,1,2,..,n$ and summation is implied over
the repeated indices. We will consider general solutions and symmetries of
these equations, and relations of these equations with the reduction of
higher-order PDE. Some new results that were needed for the comprehensive
presentation are also adduced. In particular, we will also consider discrete
symmetries of the eikonal equations equivalence classes of solutions and
relations of the symmetry solutions and the general solutions. We describe in
detail the procedure that allowed obtaining of the general solution using
hodograph and contact transformations of the initial equations. Some new
results by the author that were needed for comprehensive presentation are also
adduced - in particular, a general solution of the eikonal equation for an
arbitrary number of space variables. The paper is intended both as reference
material and as a manual.
