The standard (Kolmogorovian) notion of conditional probability is compared with the (de Finettian) notion based on the coherence principle. Each notion has both merits and drawbacks, and the talk aims to highlight them with special attention to the connection points. Various results, both old and new, are discussed and some open problems are mentioned. The talk is split into four parts. (i) Classical (Kolmogorovian) conditional probabilities; (ii) Disintegrability; (iii) Coherent (de Finettian) conditional probabilities; (iv) Some statistical implications.