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The Shadow is a very broad and abstract term that seems to be used in reference to the Darkness, the extent of evil and its influence to the minds of people as a result of the Marring of Arda. Because Morgoth tainted Arda with his rebellious spirit, his will could continue to affect the hearts of people living on Earth after he was gone. At different points the Shadow refers to the Darkness of the Enemy, Sauron or their servants, the Ringwraiths, also called Shadows (and their Black Breath). Sauron was called the Shadow of the Forest (while he dwelt in Dol Guldur) or the Shadow of the East (in Mordor). Shadow also described the Unlight of Ungoliant.
Peoples came under the sway of this corrupting power were said to be "under the Shadow". The term is also a euphemism that alludes to the seduction of Men by Morgoth in the beginning of their history: "Men came forth in the East of the world; but they fell under the shadow of the Enemy". This usage is similar to the other examples above, but is more far-reaching in that all Men living at that time were said to have succumbed to Morgoth's power. The Men fell under the Shadow already in Hildorien and became afraid of the Gift of Men. A similar fear came in the realm of Númenor when its people became restless and began to envy the immortality of the Elves.
A few kindreds of Dwarves also fell under the Shadow, especially among those clans that lived in the mountains to the far east of Middle-earth. Some Dwarves fought on Sauron's side during the Last Alliance.
In more general applications, "Shadow" is a term that apparently refers to the bewilderment and darkness accompanying certain enchantments. Sauron's tower of Barad-dûr was surrounded by Shadow, and the word is also associated with the land of Lothlórien and with the Ents.