A computability structure on a metric space is a set of sequences with certain properties that enables us to introduce classical concepts of computability theory into an arbitrary metric space. In the dissertation, special emphasis will be on computability structures that are maximal with respect to inclusion and computability structures that contain a dense sequence, i.e. separable computability structures. Any separable computability structure is maximal, but the converse is in general not true. In the dissertation, maximal computability structures on Euclidean space \(\mathbb{R}^n\) will be described precisely and it will be proved that any maximal computability structure on a subset of \(\mathbb{R}^n\) is actually an isometric image of the computability structure which consists of computable sequences in \(\mathbb{R}^n\), i.e. canonical computability structure. From that bond between maximal and canonical computability structures, it will follow that any maximal computability structure on \(\mathbb{R}^n\) is separable. We will also focus on finding the conditions under which a separable computability structure on a metric space is unique or unique up to an isometry. In doing so, we will limit ourselves to effectively compact metric spaces. A general question is whether effective compactness of a metric space implies its computable categoricity. It is known that this implication is true for metric spaces which have only finitely many isometries. In the dissertation we will expand that result with the fact that if two effective separating sequences share a computable set which has the property that there are only finitely many isometries of the underlying space which fix that set, then the sequences have to be equivalent. Moreover, we will prove that the orbit of a computable point under the isometry group is a co-computably enumerable set and we will use this to prove that certain effectively compact subsets of Euclidean space are computably categorical. From these results, it will follow that in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) effective compactness implies computable categoricity. Lastly, we will examine spaces which are effectively compact if they contain at least one effective separating sequence, i.e. categorically effectively compact spaces. We will show how the question of categorical effective compactness is connected to the question whether computable enumarability of a set implies its computability and we will show categorical effective compactness for boundaries of open bounded convex subsets of \(\mathbb{R}^n\).