The standard formalism of quantum theory is enhanced and definite meaning is given to the concepts of experiment, measurement and event. Within this approach one obtains a uniquely defined piecewise deterministic algorithm generating quantum jumps, classical events and histories of single quantum objects. The wave-function Monte Carlo method of Quantum Optics is generalized and promoted to the level of a fundamental process generating all the real events in Nature. The already worked out applications include SQUID-tank model and generalized cloud chamber model with GRW spontaneous localization as a particular case. Differences between the present approach and quantum measurement theories based on environment-induced master equations are stressed. Questions: what is classical, what is time, and what observers are addressed. Possible applications of the new approach are suggested, among them connection between the stochastic commutative geometry and Connes' noncommutative formulation of the Standard Model, as well as potential applications to the theory and practice of quantum computers.