Why does moss grow better in the shade?

Moss isn't fussy as to conditions. It needs shade--the more the better--moisture, acid and rather poor soil; exactly the opposite, in most ways, from grass. You can try transplanting moss from the woods, but the best way to acquire species that will enjoy your site is to make sure the soil is acidic, water it a bit in dry spells and wait to see what turns up. One exception is lawn moss. This is often a more sun-tolerant species and you can usually get quite a bit just by raking it out of a lawn. Try it as a mulch around flowers or shrubs.

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