Leonard Adleman of the University of Southern California initially developed this field in 1994. Adleman demonstrated a proof-of-concept use of DNA as a form of computation which solved the seven-point Hamiltonian path problem. Since the initial Adleman experiments, advances have occurred and various Turing machines have been proven to be constructible.
Since then the field has expanded into several avenues. In 1995, the idea for DNA-based memory was proposed by Eric Baum who conjectured that a vast amount of data can be stored in a tiny amount of DNA due to its ultra-high density. This expanded the horizon of DNA computing into the realm of memory technology although the in vitro demonstrations were made almost after a decade.
The field of DNA computing can be categorized as a sub-field of the broader DNA nanoscience field started by [http://seemanlab4.chem.nyu.edu/ Ned Seeman] about a decade before Len Adlemans demonstration. Neds original idea in the 1980s was to build arbitrary structures using bottom-up DNA self-assembly for applications in crystallography. However, it morphed into the field of structural DNA self-assembly which as of 2020 is extremely sophisticated. Self-assembled structure from a few nanometers tall all the way up to several tens of micrometers in size have been demonstrated in 2018.
In 1994, Prof. Seemans group demonstrated early DNA lattice structures using a small set of DNA components. While the demonstration by Adleman showed the possibility of DNA-based computers, the DNA design was trivial because as the number of nodes in a graph grows, the number of DNA components required in Adlemans implementation would grow exponentially. Therefore, computer scientists and biochemists started exploring tile-assembly where the goal was to use a small set of DNA strands as tiles to perform arbitrary computations upon growth. Other avenues that were theoretically explored in the late 90's include DNA-based security and cryptography, computational capacity of DNA systems, DNA memories and disks, and DNA-based robotics.
Before 2002, Lila Kari showed that the DNA operations performed by genetic recombination in some organisms are Turing complete.
In 2003, [https://users.cs.duke.edu/~reif/ John Reif's group] first demonstrated the idea of a DNA-based walker that traversed along a track similar to a line follower robot. They used molecular biology as a source of energy for the walker. Since this first demonstration, a wide variety of DNA-based walkers have been demonstrated.