Let $X$ be a normal complex projective variety over $\mathbb C$. In order to define the intersection product of the Chow ring, one usually requires $X$ to be smooth. How to weak the smoooth assumption is an interesting question.

Intersection on Singular Varieties and Fulton's book (e.g Chapter $18$) discuss the question before.

I am interested to the case beyond divisors and curves. For instance, one can consider intersection of surfaces in a fourfold. Normality is not enough to define an intersection product in higher dimensions.

If $X$ can be embedded into a smooth variety $Y$, we can at least intersect any cycle on $X$ with the pullback of a cycle on $Y$, by transfering the intersection to $Y$.

Assume now $X$ is a Cartier divisor on a smooth projective variety $Y$. Under mild assumption on $X$ (which seems to be necessary), how to define intersection theory of cycles on $X$ using $Y$?

If the pullback is surjective on the level of Chow groups, one can apply the pullback trick.