The term "fedayeen" (Arabic فِدائيّين, singular "fedayi") means "self-sacrificers". The term is not to be confused with Mujahidin (مجاهدين) who are "those who (perform) Jihad". Some mujahidin are fedayeen, but not all. The label "fedayeen" is closely linked to the concept of resistance fighters, particularly those who wish to present themselves as, or actually are, more hard-line in their views and methods than other competing factions. fedayeen have been found everywhere from Armenia in the 1800s to Iraq in the early 2000s, and are a constant nuisance in Afghanistan. For a better take on historical groups using the moniker "fedayeen", see Noung's writeup above. This writeup focuses on the individuals, and not the movements that they are a part of.
Topics of emphasis are the religious and social motivations that drive the concept of fedayeen, and the relationship of the concepts of fedayeen and suicide bombing in the same context. The writeup will address the motivations both from the perspective of the fedayi himself, and later, the forces driving those who seek to cultivate fedayeen. This writeup will tend to be Afghanistan-centric for the purposes of historical and cultural analysis, as that is where the bulk of my experience lies.
In the broadest sense, fedayeen are suicide soldiers. Not suicide bombers, but they might as well be for all the good it does them in practice. Their intent is to go out in a blaze of glory, and specifically to be martyred in the process. More on that later.
Firstly, the religious forces motivating fedayeen are probably the most important, and is the seed of all other justification. Most people are familiar, at least vaguely, with the concept of "72 virgins in Paradise"; though somewhat controversial in discussion of the Quran and the Hadith, this crude summary has roots in mention of Houri. Specifics of the actual reward aside, the Quran itself is explicitly clear on the topic of martyrdom; Quran 3:169-174 and 22:58 deal specifically with the subject. Essentially, martyrom is to be a topic of rejoice, not lament, for the martyr's survivors. The martyr shall receive for his repayment of Allah's grace eternal life, "a great reward" (unspecified in this passage), and Allah's pleasure. There is also precedent for martyrdom enshrined in various places in the history of Islam; the most important is probably Hussain, the third Imam and Mohammad's grandson, who was martyred by the caliph Yazid around 680 CE. Over time, his martyrdom has been interpreted by various scholars and sects to have been not a passive action, but an activist, motivated martyrdom staged in the spirit of resistance and protest against unjust rule.
These religious motivations have lead to a very complex social construct surrounding the choice to declare oneself fedayi. The typical fedayi recruit is assured of family honor, personal glory, and social stature among peers on top of the established religious rewards, which may be overstated or emphasized even more than the actual texts suggest, depending on the dogmatic beliefs of both the follower and the sponsor. It is the norm, not the exception, for a sponsor (typically a warlord or power broker dealing with a corrupt madrassa - a religious school) to make large payments of cash to the fedayi's survivors, whether spouse, children, or parents; and promise a sort of pension plan for any fedayi who by the grace of Allah isn't killed in his attempt to become a martyr. In fact, it is possible to interpret justification for this type of earthly reward for miraculously unscathed fedayi with certain readings of Bukhari 1:35, continuing the now obvious trend of manipulation and extension of textual foundations for fedayi. It is also superbly difficult for a recruit to change his mind, having committed. It is typical for a fedayi to make a "martrdom tape", a video recording of his last words and prayers, adorned with various blessed garments and carrying a rifle. The closest sense that a typical Westerner can get to the shame involved with backing out after making a martyrdom tape is a man declaring himself a pacifist in boot camp after getting a hero's send-off and hometown parade for enlisting in the armed services. The shame of a wasted martyrdom announcement is only the beginning of the social stigma that one can expect for changing one's mind. It is not atypical for fedayi to be coerced, cajoled, and borderline brainwashed at a very young age, and kept "on ice" by their handlers for a few years until they are called into action by their sponsors.
The motivations for cultivating and becoming a fedayi are very different, though subtly related. In this section I will first address the practical and social advantages of commanding a contingent of fedayeen. Essentially, it is an extreme badge of prestige. Even from a Western perspective, it is easy to understand the perception of power extended by commanding a force of soldiers so motivated that they are willing to die gladly, or at least without too much hesitation, to accomplish their goal. In Afghanistan, prestige items are very important among various warlords and power brokers. In terms of both fighting ability and social clout, they are at once resource and token. Since fedayi are generally kept by handlers, and indirectly controlled, they change hands easily without ever realizing it. Their handler, typically a mullah or other quasi-religious leader, is told by warlord A that warlord B now owns the fedayi, and if warlord B decides to give orders to the fedayi, they pass through the handler. It is important for those cultivating and keeping fedayeen to both keep a tight hold on them, but also to some extent coddle them, feed their expectations, and continue the discipline and conditioning that led them to accept the mantle in the first place.
The last major section to be addressed are both the practical and motivational differences between a fedayi, and a suicide bomber. This writeup deals specifically with minority willing suicide bombers, and not the typically coerced, drugged, or mentally disabled victims of forced suicide bombing that are much more common in Afghanistan, where suicide is traditionally strictly taboo!
As for the practical differences, they're deceptively simple to state: A fedayi walks into a situation knowing he'll be killed fighting the enemy, and a suicide bomber walks into a situation knowing he'll kill himself in order to fight the enemy. They do not seem too far removed. After all, dead is dead, but while one is a more passive acceptance of death, the other is blatantly suicide - an active ending one's own life. A common Western example springs to mind - the oft-flogged in the news media idea of "suicide by cop", where a suicidal person forces police officers to shoot them. The actions of a fedayi are similar to suicide by cop - both are attempts to make an end-run around ones' own religious rules prohibiting suicide in order to die by one's own choice, without killing oneself.
As for motivational differences, some background is in order regarding the Quran and the Hadiths' take on suicide. It is blatantly prohibited by: Quran 4:29, Bukhari 2:445, 3:438, 7:567, 7:670, 8:361, and Muslim 6480 and 6485, among others. These are simply the most straightforward and accessible exhortations against suicide. There is very little debate among Islamic scholars as to subtlety or inferred exception with these verses. Furthermore, the killing of innocents, particularly women and children, is explicitly forbidden in several places, most clearly Al-Muwatta 21.8-21.11. So, how can a willing suicide bomber be convinced to go directly against so much clear-cut scripture? The answer very often lies in how one interprets the ideas of "defensive warfare" and martyrdom as they are presented, and interact, in a specific verse - Bukhari 1:35, and a few others. Very often a young man who cannot be convinced to become a suicide bomber will take up the mantle of fedayi, instead - and some suicide bombers do not understand or have never approached the distinction between the two, and will happily blow themselves up as fedayi.
It's a messy, ugly topic, but without a writeup approximately twenty times this length, this is as far as I can take you. There is a huge apparatus lurking underneath the fedayi/suicide bomber cultivation system, consisting of a twisty maze of foreign backers, insane mullahs, brainwashing, willful and deliberate scriptural ignorance, and corruption - the type of thing that, with a few decent actors, a vignette script, and a 5 million dollar budget, might win some Sundance awards.
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Lapidus, Ira M. A History of Islamic Societies. Cambridge University Press. 1988.
The Holy Qur'an, translated by M.H. Shakir. Tahrike Tarsile Qur'an, Inc. 1983.
Translation of Malik's Muwatta, `A'isha `Abdarahman at-Tarjumana and Ya`qub Johnson.
Translation of Sahih Muslim, Abdul Hamid Siddiqui.
Translation of Sahih al-Bukhari, M. Muhsin Khan.