  We study the main open parts of the Kawaguchi--Silverman Conjecture,
asserting that for a birational self-map $f$ of a smooth projective variety $X$
defined over $\overline{\mathbb Q}$, the arithmetic degree $\alpha_f(x)$ exists
and coincides with the first dynamical degree $\delta_f$ for any
$\overline{\mathbb Q}$-point $x$ of $X$ with a Zariski dense orbit. Among other
results, we show that this holds when $X$ has Kodaira dimension zero and
irregularity $q(X) \ge \dim X -1$ or $X$ is an irregular threefold (modulo one
possible exception). We also study the existence of Zariski dense orbits, with
explicit examples.
