With the pioneering work of [Pardoux and Peng,
Syst. Contr. Lett.14 (1990) 55–61; Pardoux and Peng,
Lecture Notes in Control and Information Sciences176
(1992) 200–217]. We have at our disposal
stochastic processes which solve the so-called backward stochastic
differential equations. These processes provide us with a Feynman-Kac
representation for the solutions of a class of nonlinear partial differential equations (PDEs) which appear
in many applications in the field of Mathematical Finance. Therefore there
is a great interest among both practitioners and theoreticians to develop
reliable numerical methods for their numerical resolution. In this survey, we present a number of probabilistic methods for
approximating solutions of semilinear PDEs all based on the corresponding
Feynman-Kac representation. We also include a general introduction to
backward stochastic differential equations and their connection with PDEs
and provide a generic framework that accommodates existing probabilistic
algorithms and facilitates the construction of new ones.