Let G be a connected reductive algebraic group. I will talk about a Grobner theory for multiplicity-free G-algebras, as well as a tropical geometry for subschemes in a spherical G-homogeneous space G/H. We will discuss the notions of a spherical tropical variety and a fundamental theorem of tropical geometry in this context. We also propose a definition for a spherical amoeba in G/H and talk about the principle that amoeba approaches the tropical variety. This is directly related to the (Archimedean) Cartan decomposition for G/H. A particular case of this states that “invariant factors” of a matrix (over Laurent series) are a limit of its “singular values”. This is a joint work with Chris Manon and builds on the recent work of Tassos Vogiannou.