An important question that arises when modeling is if the unknown parameters of a model can be determined from real (and sometimes noisy) data, the so-called parameter estimation problem.  A key first step is to ask which parameters can be determined given perfect data, i.e. noise-free and of any time duration required.  This is called the structural identifiability problem.  If all of the parameters can be determined from data, we say the model is identifiable.  However, if there is some subset of parameters that can take on an infinite number of values yet yield the same data, we say the model is unidentifiable. If a model is unidentifiable assuming perfect data, then it is almost certainly unidentifiable with real, noisy data, thus knowing this information a priori helps with experimental design.  We examine this question for an important class of models called linear compartmental models used in many areas, such as pharmacokinetics, physiology, cell biology, toxicology, and ecology.  We also examine a somewhat related question called indistinguishability, which examines if two distinct models can yield the same dynamics.  In this case, two models with completely different structures can be indistinguishable from an input-output perspective.  On top of this, there are questions of observability, controllability, and model selection/rejection.  For all of these questions, we will consider the underlying graphs corresponding to our models and use tools from graph theory and computational algebra to describe and analyze our models.