The topic of brown dwarf and extrasolar planet research with the GTC and the LMT was discussed during a half day workshop at Universidad Nacional Autonoma de Mexico. This paper summarizes the contributions presented at that Workshop plus additional contributions from members of the Spanish Network for extrasolar planets. Specific observational projects, which can be carried out with the planned instruments for the GTC, including first light ones, are explained in some detail. These projects are the following: our coolest neighbors, brown dwarfs in wide binaries, brown dwarfs and planetary mass objects in Orion, follow-up of transiting extrasolar planets and verification of planet candidates, the search for ultracool companions to nearby stars, brown dwarf binaries and disks around brown dwarfs.