  Gamma-ray bursts are most luminous explosions in the universe. Their ejecta
are believed to move towards Earth with a relativistic speed. The interaction
between this "relativistic jet" and a circum burst medium drives a pair of
(forward and reverse) shocks. The electrons accelerated in these shocks radiate
synchrotron emission to power the broad-band afterglow of GRBs. The external
shock theory is an elegant theory, since it invokes a limit number of model
parameters, and has well predicted spectral and temporal properties. On the
other hand, depending on many factors (e.g. the energy content, ambient density
profile, collimation of the ejecta, forward vs. reverse shock dynamics, and
synchrotron spectral regimes), there is a wide variety of the models. These
models have distinct predictions on the afterglow decaying indices, the
spectral indices, and the relations between them (the so-called "closure
relations"), which have been widely used to interpret the rich multi-wavelength
afterglow observations. This review article provides a complete reference of
all the analytical synchrotron external shock afterglow models by deriving the
temporal and spectral indices of all the models in all spectral regimes,
including some regimes that have not been published before. The review article
is designated to serve as a useful tool for afterglow observers to quickly
identify relevant models to interpret their data. The limitations of the
analytical models are reviewed, with a list of situations summarized when
numerical treatments are needed.
