Tropical geometry is the geometry of the tropical
semiring $(\mathbb{R}, \min, +)$. Tropical varieties
are polyhedral cell complexes which behave like
complex algebraic varieties. The link between
classical complex geometry and tropical geometry is
provided by amoebas, which are logarithmic images of
complex varieties.

In this talk, we begin with a combinatorially oriented
review of some fundamental concepts in tropical geometry
(aimed at a general audience), and then we turn towards
some algorithmic problems concerning the intersection of
tropical hypersurfaces in general dimension:
deciding whether this intersection is nonempty, whether
it is a tropical variety, and whether it is connected,
as well as counting the number of connected components.
We characterize the borderline between tractable and
hard computations by proving NP-hardness and #P-hardness
results even under various strong restrictions of the
input data, as well as providing polynomial time