A single gram of DNA about the size of half inch cube can hold as much information as a trillion compact disks. This field was initially developed by Leonard Adleman of the University of Southern California. In 1994, Adleman demonstrated a proof-of-concept use of DNA as form of computation which was used to solve the traveling salesman problem.
In addition, Bernhard Yurke (Bell Labs) has developed DNA motors. Since the initial Adleman experiments advances have been made, and various Turing machines has been proved to be constructable. There are works over one dimensional lengths, bidimensional tiles, and even three dimensional DNA graphs processing.
On April 28th, 2004, Ehud Shapiro and researchers at the Weizmann Institute announced in the journal Nature that they had constructed a DNA computer, coupled with an input and output module, capable of diagnosing cancerous activity within a cell, and then releasing an anti-cancer drug upon diagnosis. DNA computing is fundamentally similar to parallel computing -- we take advantage of the many different molecules of DNA to try many different possibilities at once. The number of possible solutions to a problem grows very quickly with the size of the problem, exhibiting exponential growth.
For very large problems, the amount of DNA required will be too large to be practical. Thus, DNA computing does not provide any new capabilities from the standpoint of computational complexity theory, the study of which computational problems are difficult.