Q Gospel (hypothetical biblical writings; existence unconfirmed; 8th century BCE-11th century CE)
A diagram depicting a hypothesis of Matthew and Luke's inspiration from Mark and the Q Gospels.
Status: Existence Unconfirmed
The Q Gospel is a potential source document for various books of the new testament, containing a collection of unmitigated sayings and teachings of Jesus Christ.
Its existence is alleged by various scholars of Christian Theology including B. H. Streeter, based on similarities between New Testament gospels such as Matthew and Luke, which contain minor variations on many quotes supposed to have been said by Jesus and therefore are alleged to have used as their source the same now-missing document. The theory further states that there is are two sources to the three Synoptic Gospels, a biography (Gospel of Mark) and a book of sayings (Q). The book of sayings would explain the similarities between Matthew and Luke that cannot be explained by Mark.
Mark and Q
The first explanation is that Q was a book of sayings and teachings attributed to Jesus. This book would have counteracted the reliance on actual events that Mark had. Matthew and Luke, in turn, would have written these teachings with slight variations after taking them from a primary source, like students writing essays based on the same source material.
Just one of the numerous examples would be Jesus's 'discourse on judgmentalism':
"Judge not, that ye be not judged. For with what judgment ye judge, ye shall be judged: and with what measure ye mete, it shall be measured to you again."
"Judge not, and ye shall not be judged: condemn not, and ye shall not be condemned: forgive, and ye shall be forgiven"
This explanation is problematic, however, as the extant proto-Gospel materials share very little with Matthew and Luke, and even with each other. Early Christian writings varied wildly in their interpretations of the life and teachings of Jesus. Only Matthew, Mark, and Luke share any meaningful amount of text, narrative, and interpretation.
Matthew and Luke
Another explanation is that these books i.e. Matthew and Luke, copied one from the other, as it is generally considered to be the case that the Gospel of Mark was also used as a source by these two later-written gospels. However, by cross-sourcing quotes and references from these two books, biblical scholars tend to agree that neither Matthew or Luke is dependent on the other in the way that they are both dependent on Mark.
Q is Non-Canon
Another possible explanation is that Q is a non-canonical "Gospel of the Hebrews".  This short Gospel consists primarily of sayings attributed to Jesus. The Gospel of the Hebrews is listed second among Jerome's chronology of Christian writings, immediately after Mark. However, this is problematic, as there aren't many similarities between it and Matthew and Luke.
There is also a Three-Source Hypothesis, which suggests that Q is actually a proto-Gospel that was quoted in the works of Papias, another early Christian scholar, that is often confused for being an abridged version of Matthew.  This hypothesis further suggests that Luke was written much later than is generally accepted and took material from Q, Mark, and Matthew.
There is also a Four-Source Hypothesis, which suggests that suggests that a Q Gospel is not enough to explain discrepancies between Matthew and Luke. The idea is that Matthew and Luke not only drew material from Mark and Q, but also from a proto-Matthew (possibly supported by the writings of Papias), and a proto-Luke.
An alternative to this hypothesis is that each of the three Synoptic Gospels is largely independent of each other, but draw common elements from either an Ur-Gospel or oral sources, depending on the age of the Gospels.
Probability of Recovery
It is highly unlikely that any written version of the original Q Gospel will be unearthed, especially since, as a set of teachings, it may have existed mostly or solely as part of the oral tradition, passed down by followers only verbally as many religious texts continue to be in parts of the world even to this day. This would further explain the discrepancies between the quotes in different gospels.
However, finds of this magnitude can occur, even centuries or millennia after the fact. The most notable example being the 972 'Dead Sea Scrolls', found stored in jars in a cave in Gaza in the 1940s, which included the earliest known written versions of many biblical texts as well as many (30% of the identified texts) previously unknown extra-biblical writings which cross-referenced or confirmed certain facts and dates in the canonical Bible.
Many of the dead sea scrolls were found in very poor condition, and some have still yet to be identified. 
Notable Stories Attributed to the Q Gospel
- The Beatitudes
- Love your enemies
- Golden Rule
- Judge not, lest ye be judged
- The Test of a Good Person
- The Parable of the Wise and the Foolish Builders
- The Parable of the Lost Sheep
- The Parable of the Wedding Feast
- The Parable of the Talents
- The Parable of the Leaven
- The Parable of the blind leading the blind
- The Lord's Prayer
- Expounding of the Law
- The Birds of Heaven and The Lilies in the Field