We describe asymptotic methods for the analysis of soliton stability and related long-term dynamics in nonlinear evolution equations which conserve energy. For these equations there exists a Lyapunov functional which generates stationary soliton solutions through a constrained variational principle. We show that the stability of soliton solutions is determined in many cases by a potential function given by this functional at the stationary soliton solutions. When the potential function has a local minimum in the space of the soliton parameters the soliton solutions are stable. In the opposite case, instability of the soliton solutions takes place and we investigate the structure of the eigenvalues and unstable eigenmodes through a modification of bifurcation analysis. In an extension of this analysis, we propose an asymptotic multi-scale expansion technique and derive several universal finite-dimensional asymptotic equations governing the long-term evolution of unstable solitons of different types. Using these equations, we describe typical scenarios of this instability-induced soliton dynamics and present approximate solutions for the soliton transformation.