  Motivated by work of Zhang from the early `90s, Medvedev and Scanlon
formulated the following conjecture. Let $K$ be an algebraically closed field
of characteristic $0$ and let $X$ be a quasiprojective variety defined over $K$
endowed with a dominant rational self-map $\Phi$. Then there exists a point
$\alpha\in X(K)$ with Zariski dense orbit under $\Phi$ if and only if $\Phi$
preserves no nontrivial rational fibration, i.e., there exists no non-constant
rational function $f\in K(X)$ such that $\Phi^*(f)=f$. The Medvedev-Scanlon
conjecture holds when $K$ is uncountable. The case where $K$ is countable
(e.g., $K=\overline{\mathbb{Q}}$) is much more difficult; here the conjecture
has only been proved in a small number of special cases. In this paper we show
that the Medvedev-Scanlon conjecture holds for all varieties of positive
Kodaira dimension, and explore the case of Kodaira dimension $0$. Our results
are most complete in dimension $3$.
