When certain critical conditions are exceeded, the flow of a polymer melt becomes unstable, giving rise to particular phenomena that are the subject of this review. These instabilities are a vital industrial problem, as they are the key to productivity of extrusion processes. Indeed, their onset leads to unacceptable products and is thus the upper limit of the processing conditions. They are also a fascinating scientific problem because, although studied and discussed for more than half a century, they are still not fully understood and their mechanisms are still sometimes controversial. In the present literature review, we present an overview of the studies carried out for fifty years, as well as our own opinion on the underlying mechanisms.