Methods in finding invariant subspaces?

Question

Suppose I have a linear transformation $f: \mathbb{R^n} \to \mathbb{R^n}$ and i want to find all the invariant subspaces under $f$. What is a good approach?

I know that $\mathbb{R^n}, \{0_n\}$ are the trivial subspaces and that for $dim1$ I work with eigenvectors.But, i have $2$ questions.

1. For $ dim1$, are the eigenspaces the only invariant subspaces? If not, what do I do to find others?
2. What about higher dimensions? What do I do there?

Thank you.