Tropical varieties are balanced polyhedral complexes that arise in several contexts. In Geometry, they are commonly regarded as combinatorial shadows of their algebraic counterparts; in combinatorics, they appear in the study of realizable matroids; and in optimization, they appear as parameter domains in which the optimal solution is not unique.
    In this talk, we will discuss the different equivalent definitions for tropical varieties and the different applications of tropical geometry that they entail. Special emphasis will be put applications in economics and physics. Moreover, we will discuss how tropical varieties can be computed, and, in doing so, highlight some bread-and-butter techniques of computer algebra.