We introduce combinatorial formulas of (double) Schubert and Grothendieck polynomials based on bumpless pipe dreams and give a combinatorial proof of Monk’s rule for Schubert and double Schubert polynomials using bumpless pipe dreams that generalizes Schensted’s insertion on semi-standard Young tableaux. We also give a bijection between pipe dreams and bumpless pipe dreams and discuss its canonical nature.