In the context of families of algebraic varieties, the cohomology classes vary with families in a way which induces a specific kind of flat connections. The most important case is in Picard-Fuchs theory.
wikipedia Gauss-Manin connection
Yu. I. Manin, Algebraic curves over fields with differentiation, Izv. Akad. Nauk SSSR Ser. Mat., 22:6 (1958), 737–756, pdf(in Russian)
Boris Tsygan, On the Gauss-Manin connection in cyclic homology, Methods Funct. Anal. Topology 13 (2007), no. 1, 8394.
There is a version in noncommutative setup.