  We exploit three classical characterizations of smooth genus two curves to
study their tropical and analytic counterparts. First, we provide a
combinatorial rule to determine the dual graph of each algebraic curve and the
metric structure on the associated minimal Berkovich skeleton. Our main tool is
the description of genus two curves via hyperelliptic covers of P^1 with six
branch points. Given the valuations of these six points and their differences,
our algorithm provides an explicit harmonic 2-to-1 map to a metric tree on six
leaves. Second, we use tropical modifications to produce a faithful
tropicalization in dimension three starting from a planar hyperelliptic
embedding.
  Finally, we consider the moduli space of abstract genus two tropical curves
and translate the classical Igusa invariants characterizing isomorphism classes
of genus two algebraic curves into the tropical realm. While these tropical
Igusa functions do not yield coordinates in the tropical moduli space, we
propose an alternative set of invariants that provides new length data.
