Computer simulations and in particular mesoscopic scale simulation techniques such as the one known as dissipative particle dynamics (DPD), enable researchers to study the complexities of soft materials and polymeric systems by performing in silico that can help in the interpretation of experiments and in testing analytical theories. In addition, these mesoscopic simulations allow scientists and engineers to characterize and optimize the actual experiments in a more efficient manner. We present here a complete implementation of DPD running entirely on a graphics processing unit (GPU). We discuss the design of our algorithms and the optimizations needed to fully take advantage of a GPU. We evaluate its performance, and show that it can be more than 700 times faster than a conventional implementation running on a single CPU core. Undoubtedly, due to the immense potential that the DPD technique holds, many more significant works will emerge subsequently and the number of works involving the DPD increases exponentially, as the DPD technique continually finds new application areas and even new fields, where it can be applied and exploited. To illustrate the full potential of the technique we report several applications to soft matter systems, such as polymers in solvents in equilibrium as well as under flow.