This paper analytically characterises optimal sensor placements for target localisation and tracking in 2D and 3D. Three types of sensors are considered: bearing-only, range-only and received-signal-strength. The optimal placement problems of the three sensor types are formulated as an identical parameter optimisation problem and consequently analysed in a unified framework. Recently developed frame theory is applied to the optimality analysis. We prove necessary and sufficient conditions for optimal placements in 2D and 3D. A number of important analytical properties of optimal placements are further explored. In order to verify the analytical analysis, we present a gradient control law that can numerically construct generic optimal placements.