Transcription and translation in living cells are accomplished by mechanisms, 
which are stochastic by nature. As a result, there is a serious
and intrinsically random variability within cell populations. In particular,
usually, no cell behave in a way that might be called "average". The talk
reviews experimental examples of stochasticity of the transcription process
and gene regulation, mainly in eukaryotic cells, and discusses mathematical
levels of description of resulting stochastic dynamics. This involves
denumerable-state Markov processes, stochastic differential equations 
and diffusion equations. The aim is to show how these well-known mathematical
tools can help in novel interpretation and design of biological experiments
on single cells.