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Back when I was a junior in high school, much of my chemistry-related work entailed memorizing the periodic table. Elements 110 and 111 had been discovered in Darmstadt, Germany around that time, and my science teacher made a rather large fuss about it. My classmates made a rather large fuss about having to memorize one more element. I figured that after memorizing 109, the 110th was just a drop in the bucket, but my logic was lost on them. Unfortunately, these days the pickings are slim when it comes to discovering new elements that are stable. Scientists have isolated the elements that are available to us on earth, things like helium, tin, gold, and mercury. These elements are probably tangible to you; you can imagine a helium tank or some gold-plated jewelry. They're things that hang around; they aren't going anywhere. But elements like 110 and 111 are unstable, they can only survive for tiny fractions of seconds before decomposing to elements like lead. To understand why that is, we have to look at the problem at the atomic level. Every atom contains a nucleus composed of protons and neutrons. Protons have a positive charge and neutrons have no charge. Now, think about what it must take to tightly pack a bunch of protons together. They're all positively charged so they inherently repel, as though you forced together the wrong ends of two magnets. Physicists have described a force that keeps the nucleus from breaking apart, which solves the problem. Or does it?
It turns out that as elements become heavier (bigger than about 82 protons) and more protons are packed into that tiny little nucleus, the repulsion overwhelms the force that keeps the nucleus glued together. The story isn't quite that simple, however. For one thing, the number of neutrons also has an effect on how stable the nucleus is. Physicists have estimated that there are "magic combinations" of protons and neutrons that can confer stability even on superheavy elements. (I use "stability" loosely here. If element 110 and 111 last for only thousandths of a second, a lifetime of a day or two is very stable by comparison!) It's very challenging to do any experiments to verify and refine their calculations, but a paper published in Nature last week took a significant step in this direction.
The ref: Nature 2006 442, 896-899.
By probing how Nobelium (element 102) falls apart, researchers gained a window into the nucleus's underlying structure. They were able to hone in on numbers of protons and neutrons that they believe could exist stably.
I definitely wouldn't invest in Element 111 jewelry anytime soon, though. (Take a look at the periodic table; element 111 is located below copper, silver and gold and is likely to have similar properties.)

Image from http://www.dayah.com/periodic/