Skype presentation: Using a mathematical programming lens, we will look at a parametric family of optimal design problems in depth. Two popular problems, the approximate D- and A- optimal design problems, will be highlighted as two special members and some of the results will be customised for them. First, we will present mathematical formulations, derive their duals and obtain optimality conditions. Second, we will survey different types of algorithms that are in use and discuss their convergence properties. Third, we will provide a branch and bound framework together with techniques to generate lower and upper bounds to obtain exact designs. Along the way, we will be identifying immediate research questions, as well as theoretical and computational challenges awaiting to be tackled.