Eigenvalues and Eigenvectors of a Rotation Matrix

In robotics, rigid-body orientation is often represented with rotation matrices. Consider a $3 \times 3$ rotation matrix $R$ that rotates vectors by some angle $\theta$ about a fixed axis in 3D space.

Explain what the eigenvalues and eigenvectors of $R$ represent geometrically. In particular: (a) What is special about the eigenvector associated with the real eigenvalue, and what is that eigenvalue's value? (b) What do the complex eigenvalues tell you about the rotation? (c) Why is this decomposition useful when working with robot orientation (e.g., recovering the axis and angle of a rotation)?