One of my current research interest is mainly concerned with behaviors of open, dynamical, far-from-equilibrium systems. I would like to comprehend how such systems evolve into stable, critical states where local perturbations may spread throughout the whole systems and cause larger and larger collapses or avalanches, through extremely long processes of the so-called self-organization. I especially have concern with avalanche dynamics embedded in, and scaling behaviors exhibited by, biological evolution. My advisor, Prof. X. Cai, and I have observed a different hierarchy of avalanches by introducing a new quantity, average fitness, within the frame of the Bak-Sneppen (BS) evolution model. We have investigated the critical features of the new type of avalanches, by analytic analyses and computational simulations as well. And, we have provided analytic descriptions for under-critical and over-critical avalanches, scaling function, and avalanche moments. In addition, we have generalized the BS model and made information investigation of avalanche size distribution in the BS model.