Questions of parameter estimation – that is, finding the parameter values that allow a model to best fit some data – and parameter identifiability – that is, the uniqueness of such parameter values – are often considered in settings where experiments can be repeated to gain more certainty about the data.  In this talk, however, I will consider parameter estimation and parameter identifiability in situations where available data consists of measurements from discrete time points during a single experiment.  Our motivation comes from medical settings, where data comes from a patient; such limitations in data also arise in finance, ecology, and climate, for example.  In this setting, we can try to find the best parameters to fit our limited data.  In this talk, I will first discuss the Bayesian approach to this problem and in particular the issue of selecting a non-informative prior.  Next, I will introduce a novel, alternative goal to estimating parameter values, which we refer to as a qualitative inverse problem.  The aim here is to analyze what information we can gain about a system from the available data even if we cannot estimate its parameter values precisely.  I will discuss results that allow us to determine whether a given model has the ability to fit the data, whether its parameters are identifiable, the signs of model parameters, and/or the local dynamics around system fixed points, as well as how much measurement error can be tolerated without changing the conclusions of our analysis.  I will consider various classes of model systems and will illustrate our latest results with the classic Lotka-Volterra system.