The aim of this thesis is to study the projective and curvature symmetries in non-static spacetimes. A study of non-static spherically symmetric, non-static plane symmetric, non-static cylindrically symmetric and special non-static axially symmetric spacetimes according to their proper curvature collineations (CCS) is given by using the rank of the 6 × 6 Riemann matrix and direct integration techniques. We consider the non-static spherically symmetric spacetimes to investigate proper CCS. It has been shown that when the above spacetimes admit proper CCS, they turn out to be static spherically symmetric and form an infinite dimensional vector space. In the non- static cases CCS are just Killing vector fields. In case of non-static plane symmetric spacetimes, it has been shown that when above spacetimes admit proper CCS, they form an infinite dimensional vector space. We consider the non-static cylindrically symmetric and special non-static axially symmetric spacetimes to study the proper CCS. It has been investigated that when above spacetimes admit proper CCS, they also form an infinite dimensional vector space. We consider the special non-static plane symmetric spacetimes to investigate proper projective collineations. Following an approach developed by G. Shabbir in [39], which basically consists of some algebraic and direct integration techniques to study proper projective collineations in the above spacetimes. It has been shown that when the above spacetimes admit proper projective collineations, they become a very special class of the spacelike or timelike versions of FRW K=0 model.