There is actually a very certain limit set by quantum mechanics and thermodynamics as to how fast computers can be with chips based on silicon. Already, the layer of silicon that holds the chip is some 20 atoms across!!! Can you imagine that? And what will happen if this amount is decreased? In short, quantum physics will take over and very funny things will happen, since quantum physics suggests that things change in a random way. We won't be able to determine if the electrons flow inside the wire or outside, it will be kinda random. Additionally, heat will have a negative effect in a way that it will eventually melt the chip like cheese in toaster.
Moore's Law suggests that computer speed doubles every 18 months. This law will collapse in a very short period of time due to the above facts. As a matter of fact, this law has allready begin to collapse, since computer speed does not double every 18 months...
So, what is next? What is beyond silicon chips? There have been a number of proposals: protein computers, DNA computers, optical computers, quantum computers, molecular computers. But most probably, withing the next 10 years, known technologies will be put into the market like 3 dimensional chip arrays, or parallel processors to share the load.
Quantum computers are for the most the ultimate computers, but the known technology so far is incapable to provide stable computers. The largest computation made with a quantum computer is 3x5=15. It may not seem that much, but if you consider that only 5 atoms were used to run this calculation you get the idea of how hard it is to make calculations on quantum computers.
Reading all those are nice, it is nicer though listening to theoretical physicist Michio Kaku explaining them in this video: