TEMPERATURE
and/or
PRESSURE
(Note that the changes take place in the solid state, i.e. without melting. Ideally also take place in a closed system, i.e. without gain or loss of material. However, there is usually migration of fluids - metasomatism)
The lower limit of metamorphism is set by pressures and temperatures required to change mineralogy - usually temperatures >100°C and pressures greater than those at >2000 m. (nb - the minimum thermal gradient is about 30°C/km depth).
The upper limit is set by the point at which the rock starts to melt.