Orthogonal subspace projection (OSP) has found many applications in hyperspectral data exploitation. Its effectiveness and usefulness result from implementation of two stage processes, i.e., annihilation of undesired signal sources by an OSP via inverting a matrix in the first stage followed by a matched filter to extract the desired signal source in the second stage. This paper presents a theory of recursive OSP (ROSP) for hyperspectral imaging, which performs OSP recursively without inverting undesired signature matrices. This ROSP opens up many new dimensions in extending OSP. First of all, ROSP allows OSP to implement varying signatures via a recursive equation without reinverting undesired signature matrices. Second, ROSP can be further used to derive an unsupervised ROSP (UROSP) OSP, which allows OSP to find a growing number of unknown signal sources recursively while simultaneously determining a desired number of signal sources. As a result, the commonly used automatic target generation process (ATGP) can be extended to a recursive ATGP, which can be considered as a special case of UROSP. Third, for practical applications, UROSP can be also extended in two differ ent fashions to causal process and progressive process, which give rise to causal UROSP and progressive UROSP, respectively, both of which can be easily realized in hardware implementation. Finally, UROSP provides a feasible stopping rule via a recently developed UROSP-specified virtual dimensionality.