Equations are considered of Hamilton-Jacobi type with co-invariant derivatives with respect to functionals. These equations arise, e.g., in the theory of control and differential games with hereditary information. Alongside the known difficulties for Hamilton-Jacobi equations, they possess peculiarities connected with aftereffect. The article is devoted to developing of the theory of minimax solutions for the equations considered. The corresponding definition of the minimax solution is given. Under sufficiently general assumptions, its existence, uniqueness, well-posedness, and some other properties are proved.