We present the Global Rapid Advanced Network Devoted to the Multi-messenger Addicts (GRANDMA). The network consists of 21 telescopes with both photometric and spectroscopic facilities. They are connected together thanks to a dedicated infrastructure. The network aims at coordinating the observations of large sky position estimates of transient events to enhance their follow-up and reduce the delay between the initial detection and optical confirmation. The GRANDMA programme mainly focuses on follow-up of gravitational-wave alerts to find and characterize the electromagnetic counterpart during the third observational campaign of the Advanced LIGO and Advanced Virgo detectors. But it allows for follow-up of any transient alerts involving neutrinos or gamma-ray bursts, even those with poor spatial localization. We present the different facilities, tools, and methods we developed for this network and show its efficiency using observations of LIGO/Virgo S190425z, a binary neutron star merger candidate. We furthermore report on all GRANDMA follow-up observations performed during the first six months of the LIGO-Virgo observational campaign, and we derive constraints on the kilonova properties assuming that the events' locations were imaged by our telescopes.