The Great Ape Project is a campaign to extend basic human rights and legal protection to the Great Apes (Chimpanzees, Bonobos, Gorillas, and Orangutans) on the basis that they generally share the morally significant characteristic of personhood. While there's obviously individual variation, the average Great Ape has around the same mental abilities as a human child. We extend at least partial legal recognition and protection to the latter, and to humans a lot further down the personhood spectrum (newborn babies and the severely mentally disabled, for example) - so why not to apes? So far, the GAP has succeeded in getting bans on hominid research in several countries, but now they look to be on the verge of a real success, with Spain's government announcing its intention to legislate for hominid rights:
The Spanish Socialist Party will introduce a bill in the Congress of Deputies calling for "the immediate inclusion of (simians) in the category of persons, and that they be given the moral and legal protection that currently are only enjoyed by human beings." The PSOE's justification is that humans share 98.4% of our genes with chimpanzees, 97.7% with gorillas, and 96.4% with orangutans.
According to another article, the bill will bar the enslaving of Great Apes, as well as cruel treatment. So, it's not complete legal equality (and I wouldn't expect it to be), but rather a recognition of the basic rights to life, liberty, and freedom from torture or cruel and unusual treatment.