Abstract: We begin with a brief introduction to Grobner theory and tropical geometry. Grobner bases are one the most fundamental tools in computational algebra. Tropical geometry can be described as a piecewise linear version of algebra/algebraic geometry and comes from looking at a variety from "infinity". It has many applications in different areas such as phylogenetics and optimization. We then talk about new results (joint with Chris Manon) about far extending Grobner theory concepts and doing algorithmic computations in general algebras equipped with valuations (in particular coordinate rings of varieties). In particular, this makes a direct connection between tropical geometry and recently emerged theory of Newton-Okounkov bodies. A central problem in this web of ideas is "degenerating" a given variety to a toric variety. There are many connections with other areas such as applied algebra, symplectic geometry (Hamiltonian systems) and representation theory (reductive group actions).