Big advances in our understanding of the character of symmetries in Nature

What is a symmetry? Symmetry is an important idea in mathematics and physics. In the mathematical and physical understanding of the word, the symmetry isn't just "any kind of beauty" or "aesthetically pleasing virtue" of an object – which could be imagined by someone who is really detached from the exact content of the phrase. Symmetries are transformations you can do with an object – or the history of the Universe – so that the object looks the same afterwards (in the case of the object) or the history still obeys the same laws of physics (in the case of the symmetries of the laws).