This question, like most geopolitical situations in the Middle East, has a complex history.

Simply put, when the British carved up the Middle East following the World Wars, the small Kurdish minority got the short end of the stick in terms of a state. Now, what would be the Kurdish cultural region is a part of modern day Syria, Turkey, and Iraq. The governments of Turkey and Iraq (not sure about Syria) have had a history of persecuting Kurds. Recently, before the Arab Spring and complete collapse of Saddam, countries refused to lose/cede their land to the Kurdish regional governments.

So far, the closest the Kurds have to a autonomous state is Kurdistan in Iraq, which has gained more autonomy since the first and second gulf wars.

Aside from geopolitical borders, the people of the various Kurdish sects in Turkey, Iraq, and Syria have had trouble unifying the people under a single government. The Iraqi Kurds tend to be more democratic than the Turkish Kurds who are considered by many western nations to be a terrorist organization.

Sorry for the complete stream of consciousness here. I'm on my phone at work. I'm sure someone else will best my answer here.