They berthed (tied-up) in docks just like modern ships do. We needed to show the docks in Rome in Episode 31 but we found it quite difficult to research because there are very few Roman docks left in the modern world. Thankfully, archaeologists have excavated the harbour in Ephesus, an ancient town in Turkey, and so we used their work to give us some idea of what the harbour and dockside buildings in Rome would have looked like.
The Romans were dependent on ships to bring them food and other goods from all over their Empire but the docks in Rome could only be used by quite small boats. Larger ships had to moor in a harbour about fifteen miles away from Rome at a place called Ostia. Then the goods were moved from larger ships to smaller ones which could moor at the docks in Rome. It is still possible to see remains of some of the buildings and warehouses at Ostia. We made the arched roofs in Friends and Heroes look very much like the design of those.
Click the picture to see a larger version - you might want to save it or print it too!