  The Gerber-Shiu function provides a unified framework for the evaluation of a
variety of risk quantities. Ever since its establishment, it has attracted
constantly increasing interests in actuarial science, whereas the conventional
research has been focused on finding analytical or semi-analytical solutions,
either of which is rarely available, except for limited classes of penalty
functions on rather simple risk models. In contrast to its great generality,
the Gerber-Shiu function does not seem sufficiently prevalent in practice,
largely due to a variety of difficulties in numerical approximation and
statistical inference. To enhance research activities on such implementation
aspects, we provide a comprehensive review of existing formulations and
underlying surplus processes, as well as an extensive survey of analytical,
semi-analytical and asymptotic methods for the Gerber-Shiu function, which
altogether shed fresh light on its numerical methods and statistical inference
for further developments. On the basis of an ambitious collection of 235
references, the present survey can serve as an insightful guidebook to model
and method selection from practical perspectives as well.
