May 2016 - What we call Roman science is a mixture of two different things. The first is the discoveries and inventions made by scientists working in parts of West Asia and Africa that the Romans had conquered. In these places, scientists had already been working for a long time, and they just kept on working under Roman rule (and then under Islamic rule later on). In Phoenicia, craft workers invented blown glass, and mold-made pottery and oil lamps. It's probably Phoenicians, too, who designed better sailing ships, experimenting with building ships more efficiently from the inside out rather than from the outside in. They also developed triangular sails, which we call "lateen sails" (or Latin sails), that helped ships tack into the wind, so they could sail a different direction from the way the wind was blowing.
Nearby, in Pergamum (modern Turkey), medical research also continued from before Roman times. Galen worked there in the late 100s AD, and he was the first to describe many symptoms and treatments. His medical textbook was the standard for over a thousand years.
Throughout Roman rule the best colleges stayed where they had been before: in Egypt, and in Greece at Plato's Academy and Aristotle's Lyceum. The famous University of Alexandria kept on teaching students and gathering scholars after the Romans conquered Egypt. Egyptian doctors continued to lead the world in medicine, and here the astronomer Ptolemy developed new theories about how all the planets moved around the Earth, and even if his ideas were mostly wrong, he took very careful measurements that helped later astronomers figure out the truth. Ptolemy also produced a better map of the world, though it still got a lot of things wrong.
But the Romans themselves also built up the first good schools and research institutions in Italy and western Europe. It was these Roman engineers who invented a lot of new ways to mine for metals like silver and gold and lead. (But they learned from Chinese scientists how to use mercury amalgams to do silver-plating and gold-plating.) They developed water mills as well for grinding grain. And they were the first people to really use concrete for major building projects. The use of concrete helped them to develop the dome and the barrel vault and the cross vault. They used their vaults to build aqueducts to carry fresh water to towns, and they used their engineering skills to build sewage systems to keep their towns clean and healthy.
Mathematics during the Roman Empire probably focused mainly on developing trigonometry to be able to predict the movements of the planets. Mathematicians were held back by the clumsy Roman numbers. It wasn't until the development of our modern numbers in India that mathematics really took off.
Science in Ancient Rome, by Jacqueline Harris (1998).