Oneirology is the scientific study of dreams. It's an extensive field of research that quantitatively studies the process of dreaming. It looks for correlations between dreaming and our knowledge about the functions of the brain. It also attempts to understand how the brain works during dreaming as it pertains to memory formation and mental disorders. This is not to be confused with dream analysis, which aims to uncover the meaning behind dreams.
Other odd O_ology words:
Omnibology: The study of motor buses. Which, let's be honest, is far less interesting than say, the study of motor boats.
Ombrology: The scientific study of rain. The branch of meteorology dealing with rain.
Onomatology (onomastics): The study of proper names of all kinds and the origins of names.
Orchidology: The study of orchids (thinking of you Misha Gericke!). (:
Osmology: The science of the sense of smell and the production and composition of odors.