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This is a self-contained review about ferromagnetism in the Hubbard model,which should be accessible to readers with various backgrounds who are new tothe field.  We describe Nagaoka's ferromagnetism and flat-band ferromagnetism in detail,giving all necessary backgrounds as well as complete (but elementary)mathematical proofs. By studying an intermediate model called long-rangehopping model, we also demonstrate that there is indeed a deep relation betweenthese two seemingly different approaches to ferromagnetism.  We further discuss some attempts to go beyond these approaches. We brieflydiscuss recent rigorous example of ferromagnetism in the Hubbard model whichhas neither infinitely large parameters nor completely flat bands. We givepreliminary discussions about possible experimental realizations of the(nearly-)flat-band ferromagnetism. Finally we focus on some theoreticalattempts to understand metallic ferromagnetism. We discuss three artificialone-dimensional models in which the existence of metallic ferromagnetism can beeasily proved.Comment: LaTeX2e, 72 pages, 17 epsf figures. Many minor corrections made in  March 1998. This is the final version, which will appear in Prog. Theor.  Phys. 99 (invited paper

From Nagaoka's Ferromagnetism to Flat-Band Ferromagnetism and Beyond: An Introduction to Ferromagnetism in the Hubbard Model