In Part I of the thesis I investigate the invariants of the divided powers algebra, Γ(V) under the action of the General and Special Linear groups.  In Chapter 2 I compare the additive structure of the invariants of Γ(V) with that of the invariants of the polynomial algebra.  I show that these are not isomorphic as vector spaces (and a fortiori not isomorphic as algebras) except in two cases – when dim(V) = 2 and p = 2 or 3.  This chapter also includes some dimension counting arguments most notably the case dim(V) = 2, p = 2 in Section 2.7.  These dimension counts are useful both as techniques in their own right and because they give explicit calculations which are useful in the following chapter. In Chapter 3 I describe the algebra structure of the invariant algebra in the two cases, dim(V) = 2 and p = 2 or 3 in sections 3.4 and 3.6 respectively.  In addition I describe the algebra structure of the invariants of Γ(V) under some important subgroups of GL(V) – the transvections, the symmetric subgroup and the multiplicative subgroup.  I give complete results for the transvections and F_p^x_..  For the symmetric invariants I correct a result of Joel Segal and give a complete description of Γ_p(V_p)^Σp_..  The method used in the description of Γ(V)^SL3(V2) has potential to be extended to other cases. In Part II of the thesis I compare two different methods of defining some kind of Steenrod Operations in integral cohomology.  John Hubbuck’s K-theory squares are defined on any space homotopic to a finite CW-complex with no 2-torsion.  Reg Wood’s differential operator squares are defined only on polynomial algebras.  The question is whether considered on a suitable space these operations would be equivalent.  I show that they are essentially incompatible.