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 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.