In the last few decades the description of measurements on quantum systems has evolved into a well-established stochastic theory. This theory applies equally well to small single quantum systems subject to continuous monitoring and to quantum many-body systems where local measurements on single constituents
can now be performed in the laboratory.
In this dissertation a numerically efficient method to optimally extract and
quantify information emitted from a continuously monitored quantum system is
developed. Furthermore, we extend the concept of a quantum state, in order to
provide a framework for predicting the results of past, hidden measurements, in
the presence of both prior and subsequent measurements.
The effects of measurements on quantum many-body systems and the associated
complex dynamics is subsequently investigated. We show how to simulate the
back-action of continuous measurements on one-dimensional strongly correlated
quantum many-body systems using matrix product states. Finally, we analyse
the static and dynamic properties of a few quantum many-body models subject
to measurements and investigate the use of quantum phase transitions for high