Let M be an n-dimensional nonnegatively curved compact totally real submanifold of an n-dimensional Kähler manifold \(\bar M\) with parallel mean curvature vector. The main purpose of this paper is to classify these submanifolds when \(\bar M\) is the complex Euclidean, projective or hyperbolic space. For this, the author uses the classification of totally real parallel submanifolds in these spaces given by H. Naitoh and M. Takeuchi, and proves the following main result: ''Let M be an n- dimensional compact totally real submanifold of an n-dimensional complex space form with parallel mean curvature vector. If M has nonnegative sectional curvature, then M has parallel second fundamental form''.