Abstract: Many sensing applications are concerned with the continuous monitoring of time-dependent phenomena such as a varying electromagnetic field, temperature, or inertial effects through their influence on a quantum meter system. When a state or an observable of a single quantum system is probed continuously in time, the measurement results take the form of a time dependent measurement record, where the measurement at each instant of time is governed by Born’s rule, and the quantum meter system, in turn, evolves due to the back action of the measurements themselves. The measurement back action may benefit sensing by conditionally squeezing the measured observable and hence improving the sensitivity, and by continuously quenching the system and launching a transient response with stronger dependence on the imposed perturbations than displayed in the steady state. We review the use of quantum trajectories for optimal Bayesian estimation of imposed perturbations, the possibility to retrodict past values of a perturbation from later measurement, and theories to compute the classical and quantum Fisher information, which provide deterministic results for the sensing capabilities of a given quantum meter system.