Cosmic magnetism has that exotic "Je ne sais quoi"! Magnetism has been observed in various objects, located near the edge of the Universe and all the way down to the Milky Way's center. The observed magnetic field can take the cell-type shape in randomly-oriented large blobs found in intracluster gas or outside of clusters of galaxies, the helix shape in synchrotron jets, the longitudinal shape in ram-pressured shocks in radio lobes near elliptical galaxies, the spiral shape of logarithmic arms in spiral galaxies, or the egg shape of an enlarged interstellar bubble. In strength, the magnetic field varies from 0.1 nG (cosmological), to 20 muG (galaxies, jets, superbubbles), and to 1 mG in the Milky Way filaments. Magnetism plays a small physical role in the formation of large structures. It acts as a tracer of the dynamical histories of cosmological and intracluster events, it guides the motion of the interstellar ionised gas, and it aligns the charged dust particles. Batteries and dynamos are often employed in models to create and amplify seed magnetic fields. Starting soon after the Big Bang (redshift z > 2000), this review covers the cosmological background surface (approximate to 100, distance approximate to4.3 Gpc), the epoch of first stars (z approximate to 20; distance approximate to4.1 Gpc), the currently observable Universe (z approximate to 10, distance approximate to3.9 Gpc), superclusters of galaxies (size approximate to50 Mpc), intracluster gas (size approximate to10 Mpc), galaxies (approximate to30 kpc), spiral arms (approximate to10 kpc), interstellar superbubbles (approximate to100 pc), synchrotron filaments (approximate to10 pc), and the Milky Way's center. (C) 2004 Elsevier B.V. All rights reserved.