  We present algebraic techniques to analyze state space models in the areas of
structural identifiability, observability, and indistinguishability. While the
emphasis is on surveying existing algebraic tools for studying ODE systems, we
also present a variety of new results. In particular: on structural
identifiability, we present a method using linear algebra to find identifiable
functions of the parameters of a model for unidentifiable models. On
observability, we present techniques using Gr\"obner bases and algebraic
matroids to test algebraic observability of state space models. On
indistinguishability, we present a sufficient condition for distinguishability
using computational algebra and demonstrate testing indistinguishability.
