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Marron vs. brun

The French words marron and brun both translate to the English term brown. However, they are applied in different contexts.

Marron

A1
In French, marron denotes a specific shade of brown, which is similar to the chestnut color.
Les cheveux de Marie sont marron.
(Marie's hair is chestnut brown.)
J'ai acheté une bourse marron.
(I bought a chestnut-colored purse.)
Les feuilles deviennent marron en automne.
(The leaves turn chestnut brown in the fall.)

Brun

A1
Brun can refer to a generic brown color but it is also largely used to describe natural colors like hair, skin or eyes.
Il a les yeux bruns.
(He has brown eyes.)
La chaise est brune.
(The chair is brown.)
Mon ami est brun.
(My friend has brown hair.)

Summary

Marron and brun are both used to describe colors in the brown spectrum. However, marron usually refers to a specific chestnut brown while brun can denote a general brown color and is often used to describe natural colors like hair, eyes or skin.