Pranayama, the ancient yoga of breathing and Buteyko, the modern science of breathing are both technologies that employ the breath to develop physical and mental wellbeing and enable the practitioner to access the higher realms of consciousness. 
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The ultimate purpose of pranayama is to enter a state of trance through the cessation of all breathing.
The Buteyko method takes its name from the Russian doctor, Konstantin Pavlovich Buteyko who developed its principles in the 1950s.  It involves exercises to develop nasal-breathing, breath holding and relaxation.