This paper deals with some topics in statistical physics related to asymptotic behaviors of one dimensional diffusion type processes. More precisely, we are concerned with stochastic differential equations formally written as X(t) = b(X(t)) + σ(X(t))W(t) with a Gaussian white noise W(t), the time derivative of a Brownian motion W(t). Problems we study are 1) behaviors of X as a parameter ε → 0 which are included in b and σ; 2) behaviors of X as time t → 0 or ∞. To study these problems physicists proposed new concepts as well as methods, mostly approximate or exact only in specific situations. Since the limit of X  in case of 1) may have a singular drift or diffusion coefficient, we need to introduce a class wider than the usual diffusion processes, called bi-generalized diffusion processes. Thanks to the development of their theory, we are able to discuss those asymptotic behaviors systematically. General results of the time asymptotics 2) have also been obtained, though additional consideration is required to cover some physically interesting problems. The present paper is intended to be an introduction of these results, and is arranged so that readers may access main results by formal computation. After an exposition of basic results we will show how the general methods are applied to problems of relaxation; relaxation from a metastable state to a stable state, flip-flop processes between two stable states. As an example of time asymptotics we discuss behaviors of transition probability density and moment functions. Some of the exotic problems arising from population genetics and fracture mechanics are also discussed.