  A $\mathbf{Q}$-Cartier divisor $D$ on a projective variety $M$ is {\it almost
nup}, if $(D , C) > 0$ for every very general curve $C$ on $M$. An algebraic
variety $X$ is of {\it almost general type}, if there exists a projective
variety $M$ with only terminal singularities such that the canonical divisor
$K_M$ is almost nup and such that $M$ is birationally equivalent to $X$. We
prove that a complex algebraic variety is of almost general type if and only if
it is neither uniruled nor covered by any family of varieties being
birationally equivalent to minimal varieties with numerically trivial canonical
divisors, under the minimal model conjecture. Furthermore we prove that, for a
projective variety $X$ with only terminal singularities, $X$ is of almost
general type if and only if the canonical divisor $K_X$ is almost nup, under
the minimal model conjecture.
