  This article deals with universal deformations of dihedral representations
with a particular focus on the question when the universal deformation is
dihedral. Results are obtained in three settings: (1) representation theory,
(2) algebraic number theory, (3) modularity. As to (1), we prove that the
universal deformation is dihedral if all infinitesimal deformations are
dihedral. Concerning (2) in the setting of Galois representations of number
fields, we give sufficient conditions to ensure that the universal deformation
relatively unramified outside a finite set of primes is dihedral, and discuss
in how far these conditions are necessary. As side-results, we obtain cases of
the unramified Fontaine-Mazur conjecture, and in many cases positively answer a
question of Greenberg and Coleman on the splitting behaviour at p of p-adic
Galois representations attached to newforms. As to (3), we prove a modularity
theorem of the form `R=T' for parallel weight one Hilbert modular forms for
cases when the minimal universal deformation is dihedral.
