  A new class of compact K\"ahler manifolds, called special, is defined, which
are the ones having no surjective meromorphic map to an orbifold of general
type. The special manifolds are in many respect higher-dimensional
generalisations of rational and elliptic curves. For example, we show that
being rationally connected or having vanishing Kodaira dimension implies being
special. Moreover, for any compact K\"ahler $X$ we define a fibration $c_X:X\to
C(X)$, which we call its core, such that the general fibres of $c_X$ are
special, and every special subvariety of $X$ containing a general point of $X$
is contained in the corresponding fibre of $c_X$. We then conjecture and prove
in low dimensions and some cases that: 1) Special manifolds have an almost
abelian fundamental group. 2) Special manifolds are exactly the ones having a
vanishing Kobayashi pseudometric. 3) The core is a fibration of general type,
which means that so is its base $C(X)$,when equipped with its orbifold
structure coming from the multiple fibres of $c_X$. 4) The Kobayashi
pseudometric of $X$ is obtained as the pull-back of the orbifold Kobayashi
pseudo-metric on $C(X)$, which is a metric outside some proper algebraic
subset. 5) If $X$ is projective,defined over some finitely generated (over
$\Bbb Q$) subfield $K$ of the complex number field, the set of $K$-rational
points of $X$ is mapped by the core into a proper algebraic subset of $C(X)$.
These two last conjectures are the natural generalisations to arbitrary $X$ of
Lang's conjectures formulated when $X$ is of general type.
