A central result of information theory is the Cramer-Rao bound, which relates the accuracy of physical measurements to the Fisher information. We start with a discussion of the Cramer-Rao bound and its implications on the measurement of physical observables. We show that, when applied to the path-integral description of a stochastic process, the Cramer-Rao bound turns into a universal inequality between the fluctuations of observables and their response to perturbations of the system. This fluctuation-response inequality can serve as a starting point to derive novel relations between different physical quantities; as examples, we discuss a relation between mobility and diffusion in stochastic particle transport and the family of thermodynamic uncertainty relations, which have recently received much attention. Finally, we show how the properties of Fisher information can be used to define the notion of monotonicity for stochastic dynamics.