  An Okounkov body is a convex subset in Euclidean space associated to a big
divisor on a smooth projective variety with respect to an admissible flag. In
this paper, we introduce two convex bodies associated to pseudoeffective
divisors, called the valuative Okounkov bodies and the limiting Okounkov
bodies, and show that these convex bodies reflect the asymptotic properties of
pseudoeffective divisors as in the case with big divisors. Our results extend
the works of Lazarsfeld-Musta\c{t}\u{a} and Kaveh-Khovanskii. For this purpose,
we define and study special subvarieties, called the Nakayama subvarieties and
the positive volume subvarieties, associated to pseudoeffective divisors.
