A ring is catenary if it is Noetherian and for every pair of prime ideals $\mathfrak{p} \subset \mathfrak{q}$, any maximal chain of prime ideals $$ \mathfrak{p} = \mathfrak{p} \subset \mathfrak{p}_1 \subset \cdots \subset \mathfrak{p}_n = \mathfrak{q} $$ has the same length.
What types of structures are catenary rings suppose to capture? What are some examples and counterexamples?
Examples of catenary rings:
An integral domain of (Krull) dimension $2$ is catenary, as is any finitely generated algebra over a field. A quotient of a catenary ring is catenary.
An example of a non-catenary ring is given in Bourbaki, Commutative Algebra, ch. VIII, Dimension, §1, exercise 16 (it requires exercises 14 and 15).