Dogs and wolves are genetically 99.9% identical.
The idea that the domestic dog descended from the grey wolf was originally established in 1993 using comparisons of wolf and dog mitochondrial DNA. This investigation showed that no other living animal was more closely related to the domestic dog than the grey wolf:
The domestic dog is an extremely close relative of the grey wolf, differing from it by at most 0.2% of [mitochondrial DNA].
Mitochondrial DNA is passed down by the mother (without any genetic contribution from the father) and changes only through random mutations that occur from generation to generation. Scientists use this much smaller and separate DNA sequence to estimate when populations of animals first diverged and to estimate the evolutionary relationships between organisms.
By the mid-2000s, new technology allowed scientists to map the much longer DNA sequence found on the chromosomes of wolves and dogs — the genes that actually code for physical traits and behaviors. These studies demonstrated a 99.9% similarity (technically 99.96%) between dog and grey wolf, as covered in a 2007 review paper on the topic:
In general, the domestic dog is an extremely close relative of the grey wolf, from which it differs by only ~0.04% in nuclear coding-DNA sequence, and no dog [mitochondrial DNA] sequences have been found that show closer kinship to other canid species.
This similarity is so profound that hybridization often occurs between dogs and wolves. By the biological definition of species, this would mean that the domestic dog is a subspecies of the grey wolf (canis lupus familiaris). There remains some debate, however, as to what the appropriate classification for the domestic dog would be, with some arguing that the domestic dog should be its own distinct species (canis familiaris).
These advances in genetic analysis have allowed scientists as well to pin down the probable timing of the split between these two canid lineages. A study published in 2015, which compared the whole genome sequence of dogs to a 35,000 year old Siberian wolf specimen, suggested the split likely occurred between 27,000 and 40,000 years ago, and that domestic dogs are likely themselves the descendants of a now extinct descendant of grey wolves.