We describe the results of Monte Carlo simulations for kinetics at the collapse transition of a homopolymer in a lattice model. We find the kinetic laws corresponding to the three kinetic stages of the process : R(g)(2)(t)=R(g)(2)(0)-At-7/11 at the early stage corresponding to formation and growth of locally collapsed clusters, the coarsening stage is characterized by growth of clusters according to the law S proportional to t(1/2), where S is the average number of Kuhn units per cluster, and the final relaxation stage is described by the law R(g)(2)(t)=R(g)(2)(infinity)+A(1)((1))e(-t/tau 1(1)) with tau(1)((1)) proportional to N-2 We also present preliminary results on the equilibrium properties and "collapse" transition of a random copolymer. The transition curve is determined as a function of hydrophobic bead concentration n(a). We discuss the different collapsed copolymer states as a function of the composition. At low hydrophilicity we believe the critical value of the interaction parameter is governed by the law chi(c)(n(a)) proportional to n(a)(-2/3). In the kinetics we see unusual phenomena such as the appearance of a metastable long-lived states with few clusters and nontrivial loop structure.