  We initiate a systematic study of four dimensional $\mathcal{N}=2$
superconformal field theories (SCFTs) based on the analysis of their Coulomb
branch geometries. Because these SCFTs are not uniquely characterized by their
scale-invariant Coulomb branch geometries we also need information on their
deformations. We construct all inequivalent such deformations preserving
$\mathcal{N}=2$ supersymmetry and additional physical consistency conditions in
the rank 1 case. These not only include all the ones previously predicted by
S-duality, but also 16 additional deformations satisfying all the known
$\mathcal{N}=2$ low energy consistency conditions. All but two of these
additonal deformations have recently been identified with new rank 1 SCFTs;
these identifications are briefly reviewed.
  Some novel ingredients which are important for this study include: a
discussion of RG-flows in the presence of a moduli space of vacua; a
classification of local $\mathcal{N}=2$ supersymmetry-preserving deformations
of unitary $\mathcal{N}=2$ SCFTs; and an analysis of charge normalizations and
the Dirac quantization condition on Coulomb branches.
  This paper is the first in a series of three. The second paper, 1601.00011,
gives the details of the explicit construction of the Coulomb branch geometries
discussed here, while the third, 1609.04404, discusses the computation of
central charges of the associated SCFTs.
