  This article is meant as an accessible introduction to/tutorial on the
analytical construction and numerical simulation of a class of non-standard
solitary waves termed \emph{peakompactons}. These peaked compactly supported
waves arise as solutions to nonlinear evolution equations from a hierarchy of
nonlinearly dispersive Korteweg--de Vries-type models. Peakompactons, like the
now-well-know compactons and unlike the soliton solutions of the Korteweg--de
Vries equation, have finite support, {\it i.e.}, they are of finite wavelength.
However, unlike compactons, peakompactons are also peaked, {\it i.e.}, a higher
spatial derivative suffers a jump discontinuity at the wave's crest. Here, we
construct such solutions exactly by reducing the governing partial differential
equation to a nonlinear ordinary differential equation and employing a
phase-plane analysis. A simple, but reliable, finite-difference scheme is also
designed and tested for the simulation of collisions of peakompactons. In
addition to the peakompacton class of solutions, the general physical features
of the so-called $K^\#(n,m)$ hierarchy of nonlinearly dispersive Korteweg--de
Vries-type models are discussed as well.
