The by far dominating DNA sequencing company of the late 2000's and 2010's due to having the smallest cost per base pair.
To understand how Illumina's technology works basically, watch this video: Video 155. "Illumina Sequencing by Synthesis by Illumina (2016)".
The key innovation of this method is the bridge amplification step, which produces a large amount of identical DNA strands: Figure 61. "Illustration of the bridge amplification step of Illumina's DNA sequencing.".
This step is genius because sequencing is basically a noise to signal problem, as you are trying to observe individual tiny nucleotides mixed with billions of other tiny nucleotides.
With bridge amplification, we group some of the nucleotides together, and multiply the signal millions of times for that part of the DNA.