In this dissertation we study positive solutions to a singular p-Laplacian elliptic boundary value problem on a bounded domain with smooth boundary when a positive parameter varies. Our main focus is the analysis of a challenging class of singular p-Laplacian problems. We establish the existence of a positive solution for all positive values of the parameter and the existence of at least two positive solutions for a certain explicit range of the parameter. In the Laplacian case, we also prove the uniqueness of the positive solution for large values of the parameter. We extend our existence and multiplicity results to classes of singular systems and to the case when a domain is an exterior domain. We prove our existence and multiplicity results by the method of sub and supersolutions and our uniqueness result by establishing apriori and boundary estimates. Such results are well known in the literature for the nonsingular case. In this study, we extend these results to the more difficult singular case.