Brown moved from London, where he was born, to Chicago with his family when he was two years old. His father, originally a cabinet maker, ran a hardware store but Brown had to leave school to help support his mother and three sisters. When he finally did get to college, Crane Junior, it was forced to close in 1933 for lack of funds. He eventually made it to the University of Chicago where he obtained his doctorate in 1938. Brown then worked at Wayne University, Detroit from 1943 until 1947, when he moved to Purdue University, Indiana, where he served as professor of inorganic chemistry until his retirement in 1978.

Brown has become particularly noted for his work on compounds of boron. He discovered a method of making sodium borohydride (NaBH₄), a reagent used extensively in organic chemistry for reduction. He also found a simple way of preparing diborane (B₂H₆). By reacting diborane (B₂H₆) with alkenes (unsaturated hydrocarbons containing a double bond) he produced a new class of compounds, organoboranes, which are also useful in organic chemistry. Brown has also used addition compounds of amines with boron compounds to investigate the role of steric effects in organic chemistry. He received the 1979 Nobel Prize for chemistry.