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Let U denote a finite dimensional vector space over an algebraically closed field F . In this thesis, we are concerned with rank one preservers on the r(th) symmetric product spaces r/VU and rank k preservers on the 2nd Grassmann product spaces 2/AU. The main results are as follows: (i) Let T : [formula omitted] be a rank one preserver. (a) If dim U ≥ r + 1 , then T is induced by a non-singular linear transformation on U . (This was proved by L.J. Cummings in his Ph.D. Thesis under the assumption that dim U > r + 1 and the characteristic of F is zero or greater than r .) (b) If 2 < dim U < r + 1 and the characteristic of F is zero or greater than r, then either T is induced by a non-singular linear transformation on U or [formula omitted] for some two dimensional sub-space W of U. (ii) Let [formula omitted] be a rank one preserver where r < s. If dim U ≥ s + 1 and the characteristic of F is zero or greater than s/r, then T is induced by s - r non-zero vectors of U and a non-singular linear transformation on U. (iii) Let T : [formula omitted] be a rank k preserver and char F ≠ 2. If T is non-singular or dim U = 2k or k = 2 , then T is a compound, except when dim U = 4 , in which case T may be the composite of a compound and a linear transformation induced by a correlation of the two dimensional subspaces of U.