The topic of this work is ergodicity (stochastic stability) of various types of stochastic processes. The urge for analysis of random processes exists in every area of science and real life - medicine, biology, chemistry, physics, finance, etc., as many phenomena naturally exhibit some sort of random behaviour in their movement. Mathematical models used to describe those random movements are called stochastic differential equations (SDEs). Since the solutions of SDEs often have a very complicated structure or are impossible to obtain explicitly, it is usually hard to analyse them directly. Therefore, the emphasis has been placed on analysing their long-term stability. This includes detecting their equilibria (stacionary distributions) as well as the rate at which convergence occurs. The convergence is observed with respect to some appropriate distance function. In my work the emphasis has been placed on the quantitative aspect of this problem, namely, on finding explicit bounds on the rate of the convergence with respect to two distance functions: to total variation distance and the class of Wasserstein distances (which provides convergence in some weaker sense). As most of the existing results in this area correspond to the geometric ergodicity (that is, the case when the rate of the convergence is exponential), and known conditions ensuring sub-goemetric ergodicity are far from being optimal because there are many known examples of sub-geometrically ergodic systems that do not satisfy those conditions, the focus of my work is to find sharp and general conditions in terms of coefficients of the process that will ensure sub-geometric ergodicity of a wide range of processes. The results of my research can be divided in two parts. Firstly, I will consider classical diffusion processes - Markov processes with continuous trajectories (here, the class of processes with singular diffusion coefficients will be of special interest as they were not investigated so far). The second part of the research deals with somewhat more complicated random processes - diffusion processes with Markovian switching, which are processes that, beside the continuous, diffusive one, have a second, discrete component which changes the behaviour of the process at random times. This theory is relatively new so here we still have many interesting open questions and uninvestigated phenomena that are not characteristic for classical diffusion process. Also, in both cases, I will extend the results on a class of processes with jumps.