Roman scientific achievements are mostly in the areas of medicine and engineering. The Romans invented a lot of new ways to mine for metals like silver and gold and lead. 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.
Roman subjects in Phoenicia also invented blown glass, and mold-made pottery and oil lamps were also first made in the Roman period. Roman engineers also 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.
The Romans didn't do that much work in mathematics, held back by the clumsy Roman numbers. The Roman 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.
Science in Ancient Rome, by Jacqueline Harris (1998).