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We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is 
${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is 
$0''$. We also prove that, with a bit of work, the latter result can be pushed beyond 
$\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex automorphism orbits.
As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability.
This article contributes to the general program of extending techniques and ideas of effective algebra to computable metric space theory. It is well-known that relative computable categoricity (to be defined) of a computable algebraic structure is equivalent to having a c.e. Scott family with finitely many parameters (e.g., [1]). The first main result of the article extends this characterisation to computable Polish metric spaces. The second main result illustrates that just a slight change of the definitions will give us a new notion of categoricity unseen in the countable case (to be stated formally). The second result also shows that the characterisation of computably categorical closed subspaces of 
${\Cal R}^n $ contained in [17] cannot be improved. The third main result extends the characterisation to not necessarily separable structures of cardinality κ using κ-computability.
We study the computable structure theory of linear orders of size 
$\aleph _1 $ within the framework of admissible computability theory. In particular, we characterize which of these linear orders are computably categorical.
We study the computable structure theory of linear orders of size 
$\aleph _1 $ within the framework of admissible computability theory. In particular, we study degree spectra and the successor relation.
