Given: $0 < x < 1$

Statement I is correct: $x^2 > x^3$ Since x is a fraction (less than 1), multiplying it by itself reduces its value. Thus, $x^2 > x^3$.

Example: $x = 0.5$ $x^2 = 0.25$, $x^3 = 0.125 \rightarrow x^2 > x^3$ Hence, Statement I is correct.

Statement II is incorrect: $x > \sqrt{x}$ For $0 < x < 1$, the square root of x is greater than x itself. Example: $x = 0.25 \rightarrow \sqrt{x} = 0.5 \rightarrow x < \sqrt{x}$ Hence, Statement II is incorrect.

Therefore, only Statement I is correct.