By a linear group we shall mean essentially a group of invertible matrices over a ring. Thus, we include in our class of linear groups the 'classical' geometric groups. These are the general linear group, GL[sub]n(F), the orthogonal groups, 0[sub]n (F) and the syraplectic groups Sp[sub]n(F). The normal and subnormal subgroup structure of these groups is well known and has been the subject of much investigation since the turn of the century. We study here the normal and subnormal structure of some of their infinite dimensional counterparts, namely, the infinite dimensional linear group GL(Ω,R), for arbitrary rings R, and the infinite dimensional syraplectic group Sp(Ω,R), for commutative rings R with identity. We shall see that a key role in the classification of the normal and subnormal subgroups of GL(Ω,R) and Sp(Ω,R) is played by the 'elementary' normal subgroups E(Ω,R) and ESp(Ω,R). We shall also see that, in the case of the infinite dimensional linear group, the normal subgroup structure depends very much upon the way in which R is generated as a right R-module. We shall also give a presentation for the 'elementary' subgroup E(Ω,R) when R is a division ring.