Numerical simulations are a standard tool to investigate field theories innon-perturbative regimes. Typical algorithms used to evaluate path integralsin Euclidean space rely on importance sampling methods; i.e., aprobabilistic interpretation of the Boltzmann weight eS. However, manytheories of interest suffer from the infamous sign problem: the action iscomplex and the Boltzmann weight cannot be used as a probability distribution.Complex Langevin simulations allow numerical studies of theoriesthat exhibit the sign problem, such as QCD at finite density.In this thesis, we study methods to investigate the phase diagram of QCDin the temperature{chemical potential plane, using the complex Langevinmethod. We provide results on the phase diagram for the heavy-denseapproximation of QCD (HDQCD) for three spatial volumes, using complexLangevin and the gauge cooling technique. We also present polynomialfits of the critical temperature as function of the chemical potential foreach volume. Subsequently, we discuss instabilities encountered during thisstudy, which motivated a novel technique, named Dynamic Stabilisation,which will be introduced and the theoretical ideas behind it, explained.Dynamic stabilisation was, then, used in an investigation of the dependencyof the critical chemical potential on the hopping parameter. The two previousstudies were used to guide a second examination of the HDQCD phasediagram, focussed around the phase boundary.Lastly, we present preliminary results on the phase diagram of QCD withfully dynamical quarks at high temperatures. This shows that complexLangevin, augmented with gauge cooling and dynamic stabilisation, is suitedfor investigating QCD at finite chemical potential
