This paper discusses curvature collineations in perfect fluid spacetimes within Einstein's general relativity. Using as starting point the papers of the reviewer [Gen. Relativ. Gravitation 15, 581-589 (1983; Zbl 0514.53018)] and the reviewer are and the second author [J. Math. Phys. 32, No. 10, 2848-2853 and 2854-2862 (1991)] the paper begins with a discussion of the algebraic structures of the Riemann tensor \(R_{abcd}\) necessary for proper curvature collineations to exist (A curvature collineation \(X\) satisfies \({\mathcal L}_ XR^ a_{bcd}=0\) and ``proper'' is taken as ``not affine''). Only one of these structures (called case \(A\)) is consistent with the space-time being a perfect fluid. This type is characterized locally by the fact that the equation \(R_{abcd}k^ d=0\) for the vector field \(k\) has exactly one nonzero solution (up to scalings) at each point of space-time. Assuming case \(A\) and that the nature of \(k\) (timelike, spacelike or null) is the same everywhere, it follows that \(k\) cannot be null. The authors then show that if \(k\) is timelike then it is parallel to the fluid velocity and to a proper curvature collineation and that the space- time is either the Einstein static universe (when \(k\) has zero expansion) or a generalized Friedman model (when \(k\) has nonzero expansion). The equation of state is always \(\mu+3p=0\). They also show that when \(k\) is spacelike (and assuming a proper curvature collineation exists) then \(k\) can be scaled so that it is covariantly constant and orthogonal to the fluid velocity. The equation of state is \(\mu=p\) (stiff matter). An example of such a space-time is given. As far as the reviewer can see, only the algebraic form of the Riemann tensor is used in the above result when \(k\) is timelike whereas the specific existence of a proper curvature collineation is used in the spacelike case.