What pattern are you using? Stockinette stitch (Knit one row, purl the next) always does this. A way to avoid it is to knit a garter stitch (knit every row) border. So you would knit 4- do the pattern and knit 4.
What yarn are you using? If the pattern already has a garter stitch border, and the yarn is an animal fiber, you can block the scarf and get rid of the curling.
Also, check out the "sticky" thread about stockinette rolling for more solutions.
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Yes, the pattern calls for Stockinette stitch and the yarn I am using is acrylic/rayon. I did review the sticky thread regarding the stockinette curling. I will definitely try the suggestions regarding bordering. Thank you for your quick reply.