This started off as a reply to a recent comment in another diary, but it became so long, that I thought that it was better served as its own diary entry. The answer is after the fold.
This is a difficult question to answer, since there is no one answer. Each company or organization that is developing a new design has taken a slightly different approach to improve safety.
For example, companies like Westinghouse and General Electric have taken their old designs and modified them to build in "passive safety" features. This means that the safety of the reactor does not depend on a piece of equipment working correctly. Even if all of the power goes out and all of the shutdown equipment fail, the laws of physics kick in and the reactor is able to cool itself through natural physical process (e.g., natural convection in water) without the need of any computers, pumps, valves, etc., or any action on the part of the operators.
The French and the Germans -- i.e., the company known as AREVA -- has taken the opposite approach. Their new design still relies on "active" safety systems, but it has built in additional redundancy beyond the redundancy that exists in today's nuclear plants. Each system that is capable of shutting down and cooling the reactor in the event of an emergency is called a "safety train," and the EPR (the AREVA design) has four of them. That is, it has a safety system and three backups. Two of these systems are located in hardened buildings, which are capable of surviving the impact of an aircraft (for example), and the systems are physically positioned around the reactor so that anything impacting one side of the plant cannot affect all four systems.
In addition to this, the EPR has a little-advertised feature that is colloquially known as a "core catcher." In the worst-case scenario (such as the one highly dramatized in The China Syndrome) in which the reactor core melts and burns through the bottom of the steel vessel, there is a device below the reactor that catches the "corium" (the technical term for the molten form of a nuclear reactor core) and deposits it on a large horizontal surface, where it spreads out, cools, solidifies, and poses no additional threat. All of this takes place within the reactor's double walls of containment.
The EPR is so over-engineered that even the Union of Concerned Scientists has reluctantly admitted that this is a "pretty safe" design.
So the engineers have taken something that is very safe and made it even safer. But it gets even better. New reactor concepts are even more safe (theoretically) than the designs that I have just discussed.
For example, a new concept that has received a good amount of press in the last five years is the pebble-bed concept. This type of design was pioneered by the Germans in the sixties, seventies, and eighties, and has been picked up recently by the South Africans and the Chinese. It is a promising concept that could be developed as a relatively small, proliferation-resistant reactor that is capable of being widely deployed in the near future -- say in the next 20 years or so.
Often the pebble-bed design is advertised as being "meltdown proof." Well, that's one way of looking at it, but it is not entirely accurate. In truth, the fuel can melt (hence, a "meltdown" ... oh dear), but since it is contained in little tiny spheres, about the size of a poppyseed, that provide a very tough coating of silicon carbide, the solid/liquid state of the fuel is irrelevant to safety. In fact, the real safety value of a reactor of this type is that there is such a large thermal mass from all of the graphite in the reactor core (which is used to slow down the neutrons) that the fuel particles (the little poppyseed things) don't get hot enough to cause a problem. In an accident involving this type of reactor, the core heats up, and then, a couple of days later, cools down without any serious consequences and without any significant release of radionuclides into the environment.
Other advanced designs have nifty features as well. One that is particularly popular on the Internet these days is the Molten Salt Reactor. In this design, the fuel is dissolved in a molten salt (e.g., a fluoride salt) that is passed through the reactor core. Since the fuel is already in a molten state to begin with, this design is inherently "meltdown proof."
There are many other designs that I have not covered here, but the purpose of this text was to discuss how safety has improved since the reactors that are operating today were built. Remember that today's reactors were designed in the sixties using slide rules and mainframe computers. Technology marches on and gets better all of the time. What was once safe is now even safer.