  The Painlev\'e analysis introduced by Weiss, Tabor and Carnevale (WTC) in
1983 for nonlinear partial differential equations (PDE's) is an extension of
the method initiated by Painlev\'e and Gambier at the beginning of this century
for the classification of algebraic nonlinear differential equations (ODE's)
without movable critical points. In these lectures we explain the WTC method in
its invariant version introduced by Conte in 1989 and its application to
solitonic equations in order to find algorithmically their associated
B\"acklund transformation. A lot of remarkable properties are shared by these
so-called ``integrable'' equations but they are generically no more valid for
equations modelising physical phenomema. Belonging to this second class, some
equations called ``partially integrable'' sometimes keep remnants of
integrability. In that case, the singularity analysis may also be useful for
building closed form analytic solutions, which necessarily % Conte agree with
the singularity structure of the equations. We display the privileged role
played by the Riccati equation and systems of Riccati equations which are
linearisable, as well as the importance of the Weierstrass elliptic function,
for building solitary waves or more elaborate solutions.
