The efficient use of modern supercomputers has become one of the key challenges in computational science. For the numerical solution of time-dependent processes, time-parallel methods have opened new ways to overcome both strong and weak scaling limits. If higher order accuracy in time is needed or at least feasible, parallelization techniques based on collocation methods can provide temporal parallelism within a single time-step as well as across multiple time-steps. In this talk, we give an overview of three different approaches to introduce parallel-in-time integration for collocation methods: (1) parallel preconditioners for spectral deferred corrections, (2) the parallel full approximation scheme in space and time, and (3) diagonalization-based preconditioning for multi-step collocation problems. Those approaches can even be combined to obtain multi-time-parallel integrators. We shed light on the pros and cons of the different variants, their implementation on HPC systems, as well as existing and potential applications. We also discuss current roadblocks and further research directions.