Let G be a group and let A be finite dimensional vector space over an arbitrary field. We study the structure of linear subshifts Σ⊂AG and several dynamical aspects of linear cellular automata τ:Σ→Σ. We show that if G is polycyclic-by-finite then every linear subshift Σ⊂AG is of finite type. We prove that τ is nilpotent if and only if its limit set, i.e., the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated, and Σ is topologically mixing, we show that τ is nilpotent if and only if its limit set is finite dimensional. Joint work with Michel Coornaert (Strasbourg) and Xuan Kien Phung (Strasbourg and Montreal).