They may be the last thing you put on, but scarves can instantly transform anything you wear. Take a pair of jeans and a T-shirt, add a silky swath of flowers and tassels and you’ve got a whole new way of greeting the world. Oh, yeah, they can keep you warm, too! So many possibilities, so little time. You’ll want to tie one of these scarves on this fall:

–The Classic Square. These long-time favorites are back this year in all their chic glory. Think of these little scarves as the new choker necklace. Just fold them in half to form a triangle and tie around your neck. Whether it’s a dollar store bandana or a famous Hermes silk, square scarves are here to stay. Start your collection today. You’ll have plenty of scarves to fit any mood…whether you want to ride the range with a cowboy neckerchief one day or breeze around in a convertible a la Audrey Hepburn in a glamourous headwrap.

–The Long Rectangle. One of the most versatile kinds of scarves you can buy, the newest rectangles are long and skinny. This scarf is perfect for those big, floppy bows that are now popping up in stylish circles. But in beaded, tasseled and fringed versions, it is one of the best ways to dress up an evening tunic top or sheath just by draping the ends around your neck and letting them dangle down the front. Or if you’re in more of a “I love Lucy” mood, just wrap a long skinny scarf around your forehead.

–The Infinity Loop. Pull this popular scarf over your head and go. It’s easy and in chunky crocheted knits, it can keep you cozy and warm, layered over sweaters and coats. Ditto for the “snood,” a form of neckwear touted by many designers, which is looped closely around the neck. Scooch these muffler-like scarves up over the back of your head, and you’ve got even more style…and more protection from winter chills. The latest infinity scarves have endless possibilities.

–The Wrap Scarf. Everyone should have at least one, if not more of these scarves in their accessory wardrobe. Larger in size, whether a rectangle or square, these multipurpose wraps go the distance in our wardrobes. They make a big statement, and you don’t really have to worry about what’s underneath…a white T-shirt or a black turtleneck. The scarf does all the work. That ubiquitous pashmina scarf in all its rainbow of colors, now starring at an airport kiosk near you, may do in a pinch, but this fall, there are so many more possibilities…and patterns.

–The Blanket Shawl. These are the statement-makers of fall and winter. Bigger is better. These oversized cape-like “scarves” can easily do double duty as coats. In colorful cashmere, they are collectible classics; in plaids and prints, they are trendsetting fashion accessories…and of course, they can warm you up on the coldest of days. (And they can be your best friend on an airplane, too!)

–The Fluffy Scarf. Fur is having a huge moment now, and in fluffy scarves and wraps, you can expect to see the fuzz flying this season. With so many innovative ways to manufacture the “soft” stuff now, even high-fashion designers are getting in on the act. Watch for the furry fun to show up in rich, deep jewel-tone colors that will give anything you wear a touch of luxury, whether real or faux.

–The Signature Scarf. Yes, these are the designer scarves that are works of art signed and delivered with artistic flair. From Hermes to Gucci to Dior and more, these scarves are style statements handed down by grandmothers and mothers generation after generation. If you really want to make a lifetime investment in a scarf, this is the way to do it.

Signature designer scarves never go out of style. Shown: Hermes silk square scarf designed by artist Elaine Lustig Cohen, “Centered Rhyme.” (www.usa.hermes.com)