The son of educator Samuel Silas Curry, he was educated at Harvard University and received a Ph.D. from Göttingen in 1930, under the supervision of David Hilbert.

While at Göttingen, Curry read the published version of Moses Schönfinkel's 1920 lecture introducing combinatory logic, the fateful event in his career. He then wrote his Ph.D. thesis on combinatory logic. By working in the area for his entire career, he essentially became the founder and biggest name in the field. Combinatory logic is the foundation for one style of functional programming language. The power and scope of combinatory logic is quite similar to that of the lambda calculus of Alonzo Church, and the latter formalism has tended to predominate in recent decades.

He taught at Harvard, Princeton, and from 1929 to 1966, at the Pennsylvania State University. In 1942, he published Curry's paradox. In 1966 he became professor of mathematics at the Universiteit van Amsterdam.

Curry also wrote and taught mathematical logic more generally; his teaching in this area culminated in his 1963 Foundations of Mathematical Logic. His preferred philosophy of mathematics was formalism (cf. his 1951 book), following his mentor Hilbert, but his writings betray substantial philosophical curiosity and a very open mind about intuitionistic logic."