The phase transition via homogeneous nucleation is a fundamental process and plays important roles in many areas of science and technology, however, serious unrealiability remains in model predictions for nucleation rates. We recently performed direct, large molecular dynamics (MD) simulations of some homogeneous nucleation processes: vaporto-liquid nucleation with (1-8)×10⁹ Lennard-Jones (LJ) atoms (or 4×10⁶ water molecules), and liquid-to-vapor nucleation with 5×10⁸ LJ atoms. These large system sizes allow us to measure extremely low and accurate nucleation rates. Our MD simulations of argon vaporto-liquid nucleation succeeded in quantitatively reproducing the nucleation rates obtained in recent laboratory experiments at the same pressures and temperatures. It is also
possible to determine the formation free energy of clusters over a wide range of cluster sizes from measurements of the cluster size distribution and to test the nucleation theory from the precise comparisons. Our results indicate that the classical nucleation theory needs updates in the surface energy of nano-sized clusters, the sticking probability, and the prefactor in the nucleation formula.