Answer by taking a quote from the following article:

Gilliam was born in Minneapolis, Minnesota, the son of Beatrice (nee Vance) and James Hall Gilliam. His father was a travelling salesman for Folgers before becoming a carpenter. Soon after, they moved to nearby Medicine Lake, Minnesota. The family moved to the Los Angeles neighbourhood of Panorama City in 1952.

Well, I really want to encourage a kind of fantasy, a kind of magic. I love the term magic realism, whoever invented it - I do actually like it because it says certain things. It's about expanding how you see the world. I think we live in an age where we're just hammered, hammered to think this is what the world is. Television's saying, everything's saying 'That's the world.' And it's not the world. The world is a million possible things.  As for his philosophical background in screenwriting and directing, Gilliam said on the TV show First Hand on RoundhouseTV, "There's so many film schools, so many media courses which I actually am opposed to. Because I think it's more important to be educated, to read, to learn things, because if you're gonna be in the media and if you'll have to say things, you have to know things. If you only know about cameras and 'the media', what're you gonna be talking about except cameras and the media? So it's better learning about philosophy and art and architecture [and] literature, these are the things to be concentrating on it seems to me. Then, you can fly...!"  His films are usually imaginative fantasies. His long-time co-writer Charles McKeown commented, "the theme of imagination, and the importance of imagination, to how you live and how you think and so on ... that's very much a Terry theme." Most of Gilliam's movies include plotlines that seem to occur partly or completely in the characters' imaginations, raising questions about the definition of identity and sanity. He often shows his opposition to bureaucracy and authoritarian regimes. He also distinguishes "higher" and "lower" layers of society, with a disturbing and ironic style. His movies usually feature a fight or struggle against a great power which may be an emotional situation, a human-made idol, or even the person himself, and the situations do not always end happily. There is often a dark, paranoid atmosphere and unusual characters who used to be normal members of society. His scripts feature black comedy and often end with a dark tragicomic twist.  Gilliam is fascinated with the Baroque period because of the pronounced struggle between spirituality and rationality in that era. There is often a rich baroqueness and dichotomous eclecticism about his movies, with, for instance, high-tech computer monitors equipped with low-tech magnifying lenses in Brazil and a red knight covered with flapping bits of cloth in The Fisher King. He also is given to incongruous juxtapositions of beauty and ugliness or antique and modern. Regarding Gilliam's theme of modernity's struggle between spirituality and rationality whereas the individual may become dominated by a tyrannical, soulless machinery of disenchanted society, the film critic Keith James Hamel observed a specific affinity of Gilliam's movies with the writings of the economic historian Arnold Toynbee and the sociologist Max Weber, specifically the latter's concept of the "iron cage" of rationality.

What are some examples of his fantasy films?
The Fisher King.