The first step to make the theory of stochastic diffusion processes that arise in connection with single neuron description more understandable is reviewing the deterministic leaky-integrator model. After this step the general principles of simple stochastic models are summarized which clearly reveal that two different sources of noise, intrinsic and external, can be identified. Many possible strategies of neuronal coding exist and one of these, the rate coding, for which the stochastic modeling is relevant, is pursued further. The rate coding is reflected, in experimental as well as theoretical studies, by an input-output curve and its properties are reviewed for the most common stochastic diffusion models. The results for the simplest stochastic diffusion model, the Wiener process, are presented and from them strong limitations of this model can be understood. The most common diffusion model is the Ornstein-Uhlenbeck process, which is one substantial step closer to reality since the spontaneous changes of the membrane potential are included in the model. Both these models are characterized by an additive noise. Taking into account the state dependency of the changes caused by neuronal inputs, we derive models where the noise has a multiplicative effect on the membrane depolarization. Two of these models are compared with the Wiener and Ornstein-Uhlenbeck models. How to identify the parameters of the models, which is an unavoidable task for the models verification, is investigated. The time-variable input is taken into account in the last part of the paper. An intuitive approach is stressed throughout the review.