So in my readings, I have come across the term "Buqu" in Law and Order in Sung China and Graff's Medieval Chinese Warfare 300-900. It's clear that the "class", if it can be called that, existed for a long time--since at least the Age of Fragmentation to at least the Song Dynasty.

But who they are and what they did is unclear to me. McKnight defines them as "a type of unfree retainer" who was subject to higher penalties for crimes than ordinary commoners (and conversely, when other people committed crimes against them, their sentences were reduced from normal) and frequently uses the term "slave", whereas Graff describes them as a members of a lord or general's personal army.

Of course, their role may have changed as history went on, but it seems that they were always in some position of servility. So what exactly did they do, and would they be more like slaves, or servants? And how did they differ from regular servants and commoners in terms of duties and rights?

You have the Chinese characters for Buqu?

部曲

They're sort of like serfs.

It was most prominent in the age of fragmentation, but did exist in Tang and even Song in the more remote / newly developed places. where they settlers still faced constant dangers from indignous tribes or other things.

The term buqu had military origins. In Eastern Han armies, there were units called bu 部, which were further subdivided into qu 曲. The qu were themselves subdivided into tun 屯. We know this from the Hou Hanshu chapter on official ranks, which contains the following passage:

其领军皆有部曲。大将军营五部，部校尉一人，比二千石；军司马一人，比千石。部下有曲，曲有军候一人，比六百石。曲下有屯，屯长一人，比二百石。

(《后汉书.百官志》)

So originally, buqu simply meant the soldiers under one's command.

So would serfs be the right word?

Could they be sold or exchanged? Were they tied to the land, or to the lord? Were they servants who did duties around the house, or were they tenants who paid (rent)? What duties were they given?

As I mentioned earlier, the buqu were originally soldiers, and in the Eastern Han context these were largely either professional mercenaries or 'barbarian' tribesmen rendering military service to the Han government in place of tax.

But from the late Eastern Han civil wars of the 190s onwards, the buqu were mostly refugees who sought security and sustenance from warlords and local elites, and therefore became privately-owned serfs who served their masters as soldiers in war and farmers in peacetime. They could own property and receive plots of land as gifts from their masters, but had no tax, corvee, or military service obligations to the state - instead, their obligations were to their masters. In the Age of Fragmentation, governments occasionally held censuses aimed at converting buqu to registered tax-paying peasants, but these were obviously not popular with the buqu-owners, and often not even popular with the buqu themselves.

By the late 6th century, buqu were regarded as a class above slaves but beneath commoners. They could marry other buqu and have children with the permission of their masters, and the buqu obligation was hereditary. Under the Tang legal code, buqu could be freed by their masters to become commoners, but this had to be done officially through a hand-written letter from the head of the master household and jointly signed by the master's sons, to be submitted to the local officials for approval.

In the Tang period, due to the government's reliance on first the Garrison Militia (Fubing) system and then the use of professional mercenaries, the buqu lost their military role and were primarily farming serfs on the private manors of the rich. Then, as these manors were broken up in the disorder and civil wars of the late Tang and Five Dynasties, the use of serfdom gradually changed to the use of tenant-farming. Tenant farmers were bound to their landlords by contract and not by hereditary ownership, and their obligation was the payment of rent rather than the rendering of military and agricultural services. However, even in the Song period, the tenant farmers of more peripheral areas like Hunan and Sichuan were still effectively hereditary serfs, despite being called tenants (dian 佃) rather than buqu.

For information in Chinese, see http://zh.wikipedia.org/wiki/部曲

I ran in to another term 投下 today, and did a baidu search.

Is it fair to say TouXia is same as Buqu?

The 投下户 were from a later perid, that of the Mongol empire. These were dependents or serfs who belonged to Mongol aristocrats. Some of them had similar backgrounds to the buqu of the Three Kingdoms to Tang period, i.e. as refugees or prisoners of war. But many others were given as fiefs to the various aristocrats by the ruling Mongol khan.