Phase transitions and critical phenomena are central problems in statistical physics and Monte Carlo simulation has become a standard tool. We use state of the art Monte Carlo methods to investigate a variety of challenging problems. We mainly focus on universal quantities at phase transitions such as critical exponents and universal critical amplitudes. Because of universality highly simplified models can be used and the details can be chosen to improve simulation and convergence performance. A main theoretical approach is finite sizes scaling of Monte Carlo data to estimate thermodynamic quantities. We propose to perform systematic studies using an approach based on an intersection method to estimate critical parameters. The idea is to use the expected form of the physical quantity at the transition, which is typically a sum of a few power laws of the linear system size typically including subleading scaling corrections that are very hard to calculate. The intersection method provides a systematic technique to calculate the critical parameters. Many difficulties and limitations of traditional multi parameter fit methods could potentially be significantly improved. The full potential of the intersection approach has not been widely noted or used previously and shows great promise for substantial improvement that will hopefully give access to many problems that have been too hard and where existing results are uncertain or even contradictory. We plan to systematically apply this approach to a range of problems including the lambda transition in superfluid helium, the Kosterlitz-Thouless transition, magnetic transitions, spin glass transitions, Bose glass transitions. For example, the lambda transition has the experimentally most accurately measured critical exponents of any transitions, and are known to similar accuracy from renormalization group calculations and from simulations. However the last significant digit in the simulation results disagrees with experiments and with theory. This suggests that a more accurate simulation approach is needed for the lambda transition which is the goal here. Furthermore, we will use methods developed for the spin glass problem like population annealing to investigate phase transitions and unusual glass states in quenches and equilibrium properties of glassformers and chaotic dependence of fluctuations in the boundary conditions in spin glasses.