In this paper, new characterizations for functional observability, functional detectability, and structural functional observability (SFO) are developed, and based on them, the related optimal sensor placement problems are investigated. A novel concept of modal functional observability coinciding with the notion of modal observability is proposed. This notion introduces necessary and sufficient conditions for functional observability and detectability in a unified way without resorting to system observability decomposition, and facilitates the design of a functionally observable/detectable system. Afterwards, SFO is redefined rigorously from a generic perspective, contrarily to the definition of structural observability. A complete graph-theoretic characterization for SFO is proposed. Based on these results, the problems of selecting the minimal sensors from a prior set to achieve functional observability and SFO are shown to be NP-hard. Nevertheless, supermodular set functions are established, leading to greedy heuristics that can find approximation solutions to these problems with provable guarantees in polynomial time. A closed-form solution along with a constructive procedure is also given for the unconstrained case on systems with diagonalizable state matrices. Notably, our results also yield a polynomial-time verifiable case for structural target controllability, a problem that may be hard otherwise.