RESIDUAL JULIA SETS OF RATIONAL AND TRANSCENDENTAL FUNCTIONS Patricia Domínguez*, Núria Fagella** * F.C. Físico-Matem\'aticas, B.U.A.P Av. San Claudio, Col. San Manuel C.U., Puebla Pue, 72570, M\'exico pdsoto@fcfm.buap.mx ** Núria Fagella Dept. de Mat. Aplicada i An\'alisi Universitat de Barcelona Gran Via 585, 08007, Barcelona, Spain fagella@maia.ub.es ABSTRACT The {\em residual Julia set}, denoted by $J_r(f)$, is defined to be the subset of those points of the Julia set which do not belong to the boundary of any component of the Fatou set. The points of $J_r(f)$ are called {\it buried points} of $J(f)$ and a component of $J(f)$ which is contained in $J_r(f)$ is called a {\it buried component}. In this paper we survey the most important results related with the residual Julia set for several classes of functions. We also give a new criterium to deduce the existence of buried points and, in some cases, of unbounded curves in the residual Julia set (the so called {\em Devaney hairs}). Some examples are the sine family, certain meromorphic maps constructed by surgery and the exponential family.