Some embedding statements.
A compact complex subvariety of ${\mathbb{C}}^n$ is a point. However, every compact real manifold of dimension $n$ can be realized as a submanifold of some ${\mathbb{R}}^{2n}$.
There are compact complex manifolds that cannot be embedded into complex projective space. An example most often quoted in textbooks is the Hopf manifold, which is not even Kahler. On the other hand, I heard that embedding into real projective space is not often considered in differential geometry.
