A baby bumper headguard cap, also known as a falling cap, or pudding hat, is a protective hat worn by children learning to walk, to protect their heads in case of falls.^[1]

Known as a pudding or black pudding, a version used during the early 17th century until the late 18th century was usually open at the top and featured a sausage-shaped bumper roll that circled the head like a crown. It was fastened with straps under the chin.

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  Teaching a Child to Walk, by Pieter de Hooch
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  Portrait of Christian Hendrik Jacob Pielat van Bulderen in falling cap, by Pierre Frédéric de la Croix
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  Child in a falling cap in Indonesia, 1920-1940

The modern-day version can be many colors and may cover the entire head like a helmet.