In popular culture, a unicorn is a white horse with a bony tusk, like that of a narwhal, on the front of its head. However, not all of them have this form. The two supporters of the Royal Coat of Arms of the United Kingdom (see below) are a lion and a unicorn, but this example, while being mainly horse-like, has a very long tail bearing tufts of fur and it also has cloven hooves. This is confusing to a Biologist interested in these animals, as unicorns thus have the characteristics of both perissodactyls (odd-toed ungulates, e.g. horses and rhinoceroses) and artiodactyls (even-toed ungulates, e.g. sheep, goats, deer, cattle). Although they are mythological creatures, there has always been a fascination with the possibility of finding one.
In The Romance of Natural History, Philip Henry Gosse  includes reports of sightings of unicorns from Africa and he describes a drawing made of a cave painting:
In this were represented, with exceedingly characteristic fidelity, several of the common antelopes of the country, such as a group of elands, the hartebeest, and the springbok; while among them appeared, with head and shoulders towering above the rest, an animal having the general character of a rhinoceros, but, in form, lighter than a wild bull, having an arched neck, and a long nasal horn projecting in the form of a sabre.
This unicorn was a giant rhinoceros, of a type now extinct, which co-existed with humans perhaps tens of thousands of years ago. Recently, remains of Elasmotherium sibiricum, named the "Siberian Unicorn", were found in shallow sediments in Kazakhstan. Shpansky, Aliyassova and Ilvina  were able to date the remains as being less than 30,000 years old, confirming that these giant rhinoceroses also co-existed with humans.
The horn of E. sibiricum, like that of modern rhinoceroses, is made of keratin and so, like nails, it becomes broken down over time in a way that calcified tissue does not. As a result, we have no direct record of the size and shape of the horn (as far as I know) and must depend on assumptions based on features of the skull. The skull of E sibiricum has a large protuberance that is believed to have supported the horn and the News section of Nature  reported in 1878:
..the protuberance of the skull of the elasmotherium presents a rough, uneven surface, traversed by deep furrows once occupied by blood vessels. The whole analogy with the rhinoceros points with the greatest certainty to the previous existence of a horn, which, to judge from the size of the blood-vessels once encircling the base, must have possessed enormous dimensions, and easily exceeded the length of the skull itself.
Measurements taken by Shpansky et al. on the skull they examined gives the diameter of the protuberance, and thus the base of the horn, to be at least 26 cm across and all reconstructions of the Siberian Unicorn suggest a very large horn must have been present (see below).
Our knowledge of the appearance of animals that are reconstructed from fossil material comes from artists' impressions, based on the advice of experts in imagining whole animals from fossil fragments. We love these impressions, with many new finds, perhaps based on just a few pieces of skeleton, reported regularly by the media. We go further with animatronics and video techniques, showing movement and behaviour of extinct animals within their presumed environment. These are hugely popular, but are they always accurate?
Interestingly, large and/or dangerous extinct animals hold the greatest interest for us because they induce fear, an emotion that we enjoy at a safe distance. We recognise that there is a difference between myth and reality, yet the margin between the two can become blurred, especially when we let our imaginations have free rein. Having said that, wouldn't it be marvellous to discover a herd of living E. sibiricum, so that we know what they look like and how they behave?
 Philip Henry Gosse (1860) The Romance of Natural History. London, J. Nisbet and Co.
 Andrei Valerievich Shpansky, Valentina Nurmagambetovna Aliyassova and Svetlana Anatolievna Ilvina (2016) The Quaternary Mammals from Kozhamzhar Locality (Pavlodar Region, Kazakhstan). American Journal of Applied Sciences 13: 189-199.
 News (1878) The Elasmotherium Nature 18: 387-389