To elevate or improve one's social, political, and/or financial position in life; to become more successful than one was before. You're only going to truly move up in the world if you make a point of rubbing elbows with those of a higher social standing. It's unsurprising how quickly Sarah has moved up in the world—her tenacity and determination are matched only by her intelligence and talent. The Robinsons really moved up in the world after they won the lottery.