Let $M$ be an m-dimensional real manifold, generically embedded in a complex n-dimensional manifold $(X, J)$. For a point $p \in M$ , the expected dimension of the maximal complex subspace of $T_pM$ is $m-n$. Points in $M$ where this dimension is strictly more than $m - n$ are called CR singular. We will give an overview of the well developed theory of CR singular points of real surfaces in complex surfaces and then show some more recent results about CR singular points in higher dimensions.