Here is a list of achievements and inventions that most scholars agree were invented or achieved by the ancient Romans, many of which influence our life today:
Technology: The ancient Romans were great builders. No one is arguing about that! The ancient Romans built things to last. Their advances in technology include the actual invention of concrete, roman roads, roman arches, and aqueducts.
Medicine: Huge public health programs including welfare programs for the poor
Language: Roman language (Latin), the root of all the romance languages including English, French, and Spanish
Religion: The Catholic faith, which kept learning alive after the fall of the Western Roman Empire
Roman Law/Politics, including the law that states a person is innocent until proven guilty (from the Twelve Tables), and propaganda campaigns including coinage with the current emperor's picture on it, to remind everyone who was in charge of the empire
Literature/Theatre/Art: Satire, which was a loud and rude sort of sarcastic approach to comic theatre, works of literature including Virgil's Aeneid, realistic statues, ornate jewelry, masks, mosaics.
Customs, including the use of rings to denote friendship, engagements, and weddings, and the use greenery to decorate during winter holidays, and other holiday customs
Clothing: Socks (called soccus by the ancient Romans) worn by both women and men. Shoes, all kinds of shoes, including the hobnailed shoe that made such a scary racket when worn by the common soldier - along with shoe construction that adjusted for left foot, right foot variance in shape, which made wearing any shoes a lot more comfortable. And a bunch of neat hats.
Games: Many board and ball games including knuckleball (jacks) and hoops