To prepare the mussels, scrape off anything living on the outside of the shells and pull out the dark hairs (the beard) coming out of the side of the shells. Soak them in cold water while doing so. The shells are usually closed when you buy them. Do not panic if they open under water! Any shells that do not close when you tap them have to be thrown out. This part may be done a couple of hours ahead of time if the mussels are stored in a cool place. Set the cleaned mussels aside.
In a large heavy-bottomed pot, combine the carrot, onion, celery, bay leaf, peppercorns, lemon slices and white wine. Bring to the boil over high heat and add 1/3 of the cleaned mussels. Cover the pan and cook, shaking every minute or so to help the mussels cook evenly. When all of the mussels are open, remove from the pan using a slotted spoon and start enjoying your meal. Continue to cook remaining mussels in the same liquid.