  This memoir is devoted to the study of formal-analytic arithmetic surfaces.
These are arithmetic counterparts, in the context of Arakelov geometry, of
germs of smooth complex-analytic surfaces along a projective complex curve.
Formal-analytic surfaces provide a natural framework for arithmetic
algebraization theorems, old and new.
  Formal-analytic arithmetic surfaces admit a rich geometry which parallels the
geometry of complex analytic surfaces. Notably the dichotomy between
pseudoconvexity and pseudoconcavity plays a central role in their geometry.
  Our study of formal-analytic arithmetic surfaces relies crucially on the use
of real-valued invariants. Some of these are intersection-theoretic, in the
spirit of Arakelov intersection theory. Some other invariants involve
infinite-dimensional geometry of numbers.
  Relating our new intersection-theoretic invariants to more classical
invariants of Arakelov geometry leads us to investigate a new invariant, the
Archimedean overflow, attached to an analytic map from a pointed compact
Riemann surface with boundary to a Riemann surface. It is related to the
characteristic functions of Nevanlinna theory.
  Our results on the geometry of formal-analytic arithmetic surfaces admit
applications to concrete problems of arithmetic geometry. Notably we generalize
the arithmetic holonomicity theorem of Calegari-Dimitrov-Tang regarding the
dimension of spaces of power series with integral coefficients satisfying some
convergence conditions.
  We also establish an arithmetic counterpart of theorems of Lefschetz and Nori
by providing a bound on the index, in the \'etale fundamental group of an
arithmetic surface, of the closed subgroup generated by the \'etale fundamental
groups of some arithmetic curve and of some compact Riemann surfaces mapping to
the arithmetic surface.
