The magnetic phase diagram of high-temperature superconductors can contain many exotic vortex phases not observed in conventional superconducting materials. For example, the familiar vortex lattice may melt at high temperatures into a vortex liquid. The influence of defects, which pin the vortices, is of particular interest from both a theoretical and an experimental point of view. We have used a combination of small angle neutron scattering (SANS) and muon-spin rotation to probe the order of the vortex system on a microscopic scale and have succeeded, for the first time, to measure a well-ordered vortex lattice (VL) structure at all doping regimes of LSCO. In the optimally to overdoped regime a field-induced transition from hexagonal to square coordination is reported. The possible connections of our neutron results to photoemission data, as well as the implications for various competing theoretical models will be discussed. In the underdoped regime we observe, as a function of applied magnetic field, a transition from an ordered vortex state to a vortex glass phase that results from the presence of random pinning. Finally, recent measurements of the VL on electron doped high-temperature superconductors are presented.