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Burgundy is one of those hues that looks great on every skin tone and makes a chic color choice when choosing a dress to wear to any occasion. You will, of course, need to add hosiery to your look if the weather is on the cold side. Don't fret though, because you should be thinking of your tights as a fashion accessory and not a fashion burden. With lots of color options, you'll be able to create multiple ensembles with the same dress.
Black tights are simple, and they come in more styles than you might think. Basic black opaque tights are perfect for your burgundy dress, but you can also go with semi-transparent black tights, too. With a young dress and accessories, transparent tights don't look "granny" at all. If you want an even more interesting version of black tights, try printed or embroidered ones with floral, striped or other fun patterns. Because these are still basic black, they'll add visual interest, not clash with or detract from your burgundy dress.
Gray tights are an appropriate option to don under a burgundy dress. Both are great fall or winter colors, and they complement each other well -- yet gray will leave the emphasis on your dress. Try wearing various textures in this color, too, like wool or gray prints; and don't be afraid to go light! The "Chicago Tribune" notes that "wearing lighter-colored tights such as a dove gray ... will introduce a romantic element to your outfit." Who doesn't want a little romance?
To try something different but remain in a neutral palette, try navy blue tights with your burgundy dress. This is a bolder choice; depending on how obviously blue your hosiery is, it can be relatively fashion forward but still totally doable. With this type of color-blocking, try to keep the rest of your ensemble somewhat neutral or go with variations of the hues you're already wearing.
Multicolor tights in insane color combinations -- like those worn by clowns -- exist, but they are not hosiery that would match and enhance your burgundy dress ensemble. Instead, try one of the myriad tights that would look fabulous with your burgundy dress and contain more than one relatively mild hue. For example, gray tights with colored embellishments like small hearts or flowers would work, as would black and white tights in a subtle pattern. You don't have to stick to completely neutral colors, but make sure your tights aren't so busy that you end up looking like a member of the circus.