Part 1: Asymptotic nilpotency of classical cellular automata A cellular automaton (CA) is a continuous shift-commuting self-map of a subshift. By a classical CA we refer to a CA defined on a full shift over a free abelian group, namely on $S^{\mathbb{Z}^d}$ for a finite alphabet $S$ under the shift action of $\mathbb{Z}^d$. For $d = 1$, it was shown in [Guillon-Richard, 2008] that if such a CA is asymptotically nilpotent (every configuration converges to the all-zero configuration), then actually every point maps to the all-zero point after a bounded number of steps. This was generalized to CA on free abelian groups by the author in 2012. We explain some of the geometric ideas behind the proofs, and discuss the importance of nilpotency in the decidability theory of CA. $$ \, $$ Part 2: Asymptotic nilpotency of non-classical cellular automata We call a (topological dynamical) $\mathbb{N}$-system asymptotically nilpotent (AN) if there is a special point 0 such that every point converges to 0 in the iteration of the dynamics, and uniformly asymptotically nilpotent (UAN) if this convergence is uniform over the space. A non-uniformly asymptotically nilpotent (NUAN) $\mathbb{N}$-system is one that is AN but not UAN. We say a family of $\mathbb{N}$-systems is nilrigid if it does not contain a NUAN map, and if $G$ is a group, we say a $G$-system is nilrigid if its family of endomorphisms ($G$-commuting continuous self-maps) is nilrigid. The results above state essentially that full shifts over abelian groups are nilrigid. We discuss how robust/fragile this result is, by studying nilrigidity when the "full shift" assumption is replaced by dynamical assumptions, e.g. by SFTness or soficness, when the alphabet is replaced by a manifold, or when we step into the measurable category, finding plenty of non-nilrigid situations. We then discuss the case when the acting group is non-abelian, e.g. the Heisenberg group or a free group.