  In this paper we introduce, via a Phragmen-Lindel\"of type theorem, a maximal
plurisubharmonic function in a strongly pseudoconvex domain. We call such a
function the {\sl pluricomplex Poisson kernel} because it shares many
properties with the classical Poisson kernel of the unit disc. In particular,
we show that such a function is continuous, it is zero on the boundary except
at one boundary point where it has a non-tangential simple pole, and reproduces
pluriharmonic functions. We also use such a function to obtain a new
"intrinsic" version of the classical Julia's Lemma and
Julia-Wolff-Carath\'eodory Theorem.
