Symmetry as a concept is a very integral part of our lives. Humans and animals tend towards symmetry. A lot of the things that we design are always almost symmetric. Asymmetry does exist; when it does, it's got a special place in the universe. Not all things are symmetric, but most of the world tends to follow symmetry.
The notion of beauty is based on symmetry. The definition of beauty varies; some people like chubby faces while others like strong facial features. People interpret symmetry of facial features as beauty.
Most architectural designs follow symmetry. Think of the most iconic buildings in the world, and they will exhibit some symmetry. The Taj Mahal is symmetric and so is the Eiffel Tower. Ancient civilizations built symmetric structures too. The Pyramids of Giza are one example. I can't think of anything else right now but I'm certain that any structure built by ancient civilizations would have exhibited symmetry in some form or another.
Cars and motorcycles show some symmetry. Almost all forms of transport are symmetric along one plane. Symmetry in automobiles is a necessity along the longitudinal plane. If they weren't stable along the longitudinal plane, they would be extremely unstable in motion. Engineers try to incorporate symmetry when it comes to the weight distribution of a vehicle, as it lends stability to an automobile.
Stabilty and symmetry are correlated at the micro level too. Electrons in molecules and atoms like pairing up. They always exist in pairs of opposite spin. A lone electron in an atom decreases the stability of the atom whereas all electrons paired up increases stability. All noble gases have their electrons paired up and don't react. This doesn't mean that they will not react; it just takes a lot of effort to get them to form compounds. All atoms which contain an unpaired electron will always try to achieve stability; they will find other atoms which can help pair this electron up. This is the basic rule; things aren't this simple though.
Optical Isomerism is an interesting topic. Isomers (in chemistry) are different forms of the same compound or element. Optical Isomers exist in certain compounds; they are non-superimposable mirror images of each other. Although it seems like a minor difference; the implications of this phenomenon are major. Some drugs that are used to treat medical conditions require just one of the optical isomers of the chiral compound. The other form is useless because of its spatial configuration. A right hand will only fit into a right handed glove.
Symmetry and asymmetry exist together. The earth is symmetric, more or less; the details of it aren't. So is the human body. Atoms show some form of symmetry, automobiles aren't symmetric along all planes. The BMW S1000RR is mostly symmetric, but I find the asymmetric front very attractive.
Symmetry isn't always good, and it isn't always bad. One may think of symmetry and asymmetry as good and evil. They both exist together.
If everything's good; there is no bad. Likewise, if everything is symmetric, there is no asymmetry to spice things up. They're both an essential part of our lives, and that's quite remarkable.
