  We deal with various splitting methods in algebraic logic. The word
`splitting' refers to splitting some of the atoms in a given relation or
cylindric algebra each into one or more subatoms obtaining a bigger algebra,
where the number of subatoms obtained after splitting is adjusted for a certain
combinatorial purpose. This number (of subatoms) can be an infinite cardinal.
The idea originates with Leon Henkin. Splitting methods existing in a scattered
form in the literature, possibly under different names, proved useful in
obtaining (negative) results on non-atom canonicity, non-finite
axiomatizability and non-first order definability for various classes of
relation and cylindric algebras. In a unified framework, we give several known
and new examples of each. Our framework covers Monk's splitting, Andr\'eka's
splitting, and, also, so-called blow up and blur constructions involving
splitting (atoms) in finite Monk-like algebras and rainbow algebras.
