The overall goal of this project is to analyze symmetry-breaking, energy transport and synchronization phenomena in oscillator networks with active loss and gain. In particular, I want to investigate these eects at the microscopic level, where thermal and quantum noise have a dominant inuence. Such a scenario occurs, for example, in coupled nano- and optomechanical systems, but can be implemented as well with trapped ion systems, superconducting circuits, etc. In this abstract I will summarize the three main projects of my PhD thesis.The phenomenon of breaking of the parity and time-reversal symmetry (PT -symmetry breaking) in classical systems with balanced loss and gain is associated with a sharp transition from a purely real to a complexeigenvalue spectrum of the underlying dynamical matrix. Over the past years, this phenomenon has been extensively studied, both theoretically and in experiments. Many setups used coupled optical modes where,however, the system is always in a large amplitude classical state. In this project I want to perform the - to my knowledge - rst study on the eect of PT -symmetry breaking in the low amplitude quantum regime where theeects of non-linearities and intrinsic quantum noise become important. As a model system for the theory, I analyze the stationary states of two coupled harmonic oscillators with engineered loss and gain. To do so, I need to apply dierent numerical techniques to solve the corresponding master equation for this system. Preliminary results already reveal an unconventional transition from a high-noise symmetric state to a parity-broken lasing state with strongly reduced fluctuations.Furthermore, I want to extend the analysis of PT -symmetry breaking to nite dimensional quantum systems, namely spin systems. Therefore, advanced numerical techniques for the simulation of extended PT -symmetric spin chains need to be developed in order to study a crossover from a symmetric to a symmetry-broken phase also for nite dimensional quantum systems. PT -symmetric systems containing regions with loss and regions with gain will naturally exhibit a ow of energy between the dierent domains. I want to address the quite general question of how energy is distributed in networks with gain and loss and how this symmetry breaking transition will aect energy transport properties at the micro-scale. In view of the rapid development of nanotechnologies, understanding energy transport at the microscopic level has become even more important. In this project I also want to go beyond heat transport and address the problem of active energy transport through a microscopic oscillator network. `Active' means that energy is injected by a microscopic `generator' at one end of the network and extracted by another microscopic `user' at the other end. Such a system can, for example, be realized by an array of coupled nanomechanical oscillators using optomechanical actuation and cooling techniques to implement a controlled amount of energy gain and energy loss. Due to non-linear saturation eects in the gain and loss processes, such a network can show a highly non-trivial behavior, such as a symmetry-breaking phase transition. I will evaluate the steady state energy current through this chain under the in uence of thermal and quantum noise. The results may give important insights for the forced energy transfer in microscopic networks. One key feature of coupled non-linear systems is the phenomenon of synchronization. Non-linear gain-loss systems, such as optomechanical systems can therefore be seen as suitable to study synchronization eects as one crosses from the classical to the quantum regime. In recent years, there have been several works analyzing synchronization phenomena but they usually only consider the synchronization of a gain oscillator to an external force or the mutual synchronization of multiple gain oscillators. However, in my study of PT -symmetric oscillator systems, I have discovered a surprisingly strong synchronization eect between gain and loss oscillators. Since in this case there is an active cooling mechanism present, the synchronization eect can be much more robust with respect to thermal and quantum noise, as is relevant, for example, for coupled nanomechanical oscillators or optomechanical systems. For a full understanding a detailed theoretical investigation of such synchronization eects is required as well as an analysis of potential applications.