Clay Buckets offer an alternative to the usual Iron Buckets. They're easier to craft but they don't hold up to heat quite as well so they'll break once you place Lava with them. However, they can place Water indefinitely.

To craft Clay Buckets, you first must craft an Unfired Clay Bucket, which requires three Clay Balls.

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Once you have an Unfired Clay Bucket, simply smelt it in a Furnace to obtain a Clay Bucket.
 When used to collect Water or Lava, a Clay Bucket will glow to indicate there is something inside it. You can hover over it's name to see if it contains Water or Lava.

You can use a Clay Bucket to pick up the water from a Composter or the lava from a Fresnel Lens.

Clay Buckets are mainly designed as a way to get started in the Hard Mode island and, as such, do not stack.