Existential rules, i.e. Datalog extended with existential quantifiers in rule heads, are currently studied under a variety of names such as Datalog+/–, ∀∃-rules, and tuple-generating dependencies. The renewed interest in this formalism is fuelled by a wealth of recently discovered language fragments for which query answering is decidable. This paper extends and consolidates two of the main approaches in this field – acyclicity and guardedness – by providing (1) complexity-preserving generalisations of weakly acyclic and weakly (frontier-)guarded rules, and (2) a novel formalism of glut (frontier-)guarded rules that subsumes both. This builds on an insight that acyclicity can be used to extend any existential rule language while retaining decidability. Besides decidability, combined query complexities are established in all cases.