  We are interested in two classes of varieties with group action, namely toric
varieties and spherical embeddings. They are classified by combinatorial
objects, called fans in the toric setting, and colored fans in the spherical
setting. We characterize those combinatorial objects corresponding to varieties
defined over an arbitrary field $k$. Then we provide some situations where
toric varieties over $k$ are classified by Galois-stable fans, and spherical
embeddings over $k$ by Galois-stable colored fans. Moreover, we construct an
example of a smooth toric variety under a 3-dimensional nonsplit torus over $k$
whose fan is Galois-stable but which admits no $k$-form. In the spherical
setting, we offer an example of a spherical homogeneous space $X_0$ over $\mr$
of rank 2 under the action of SU(2,1) and a smooth embedding of $X_0$ whose fan
is Galois-stable but which admits no $\mr$-form.
