Given a compact geodesic space X, we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of X to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and close, Rips and Čech induced persistences. Amongst other results we prove that a Rips critical point c corresponds to an isometrically embedded circle of length 3c, that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base, and that Rips and Čech induced persistences are isomorphic up to a factor 3/4. The theory describes geometric properties of the underlying space encoded and extractable from persistence. ŽIGA VIRK 11.1. Persistence of compact geodesic spaces for filtrations by closed complexes 31 11.2. Application: metric graphs 33 11.3. Generalizations to length spaces and other relaxations 33 12. Discussion and future work 33 References 34