  The Gemini Multi-conjugate adaptive optics System (GeMS) at the Gemini South
telescope in Cerro Pach{\'o}n is the first sodium-based multi-Laser Guide Star
(LGS) adaptive optics system. It uses five LGSs and two deformable mirrors to
measure and compensate for atmospheric distortions. The GeMS project started in
1999, and saw first light in 2011. It is now in regular operation, producing
images close to the diffraction limit in the near infrared, with uniform
quality over a field of view of two square arcminutes. The present paper (I) is
the first one in a two-paper review of GeMS. It describes the system, explains
why and how it was built, discusses the design choices and trade-offs, and
presents the main issues encountered during the course of the project. Finally,
we briefly present the results of the system first light.
