We will discuss various types of approximations to one real number by rationals. We describe different irrationality measure functions and the related Diophantine spectra, such as Dirichlet spectrum, spectra for the second best approximations, Minkowski’s spectrum and resent results on the topic. Surprisingly there are unsolved problems related to some of these objects and for some of them there is no ideas how to solve them. At the end we will say a few words about multidimensional generalisations.