  In this document we review a geometric technique, called \emph{the blow-up
method}, as it has been used to analyze and understand the dynamics of
fast-slow systems around non-hyperbolic points. The blow-up method, having its
origins in algebraic geometry, was introduced in 1996 to the study of fast-slow
systems in the seminal work by Dumortier and Roussarie
\cite{dumortier1996canard}, whose aim was to give a geometric approach and
interpretation of canards in the van der Pol oscillator. Following
\cite{dumortier1996canard}, many efforts have been performed to expand the
capabilities of the method and to use it in a wide range of scenarios. Our goal
is to present in a concise and compact form those results that, based on the
blow-up method, are now the foundation of the geometric theory of fast-slow
systems with non-hyperbolic singularities. We cover fold points due to their
great importance in the theory of fast-slow systems as one of the main topics.
Furthermore, we also present several other singularities such as Hopf,
pitchfork, transcritical, cusp, and Bogdanov-Takens, in which the blow-up
method has been proved to be extremely useful. Finally, we survey further
directions as well as examples of specific applied models, where the blow-up
method has been used successfully.
