Identities and congruences involving the Fubini polynomials

In this paper, we investigate the umbral representation of the Fubini polynomials $F_x^n:=F_n(x)$ to derive some properties involving these polynomials. For any prime number $p$ and any polynomial $f$ with integer coefficients, we show $(f(F_x){)}^p\equiv f(F_x)$ and we give other curious congruences.