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Sunflowers are ready to harvest once the heads become brown. If you have a particularly wet season however, they can go mouldy [if this is the case you will need to cut off the head once the back of it is yellowing and allow the drying process to continue in a greenhouse or shed]. You should prepare them for the drying process once the back of the head begins to turn yellow to yellow-brown.  In order to harvest the seeds, the sunflower head must be completely dry. Otherwise, the flower will not surrender the seeds. A sunflower will reach this state naturally after a few days of beginning to wilt. It is easier to dry sunflowers on the stem if you have dry, sunny weather. If you live in a moist climate, you may want to consider drying them off the stem, instead. At least half of the yellow petals should have dropped before you start preparing the sunflower for harvest. The head of the flower should also begin to droop. It may look dead, but if it still has its seeds, then the sunflower is drying out in a healthy manner. Examine the seeds. Even if they are still stuck on the flower head, they should begin to plump up. The seeds should also be hard and may have their trademark black-and-white striped shell, or purely black, depending on the type of sunflower. Cover the head of the flower with a paper bag, tying the bag on loosely with twine or yarn to prevent it from being knocked off.  You could also use cheesecloth or similarly breathable fabric, but you should never use a plastic bag. Plastic will restrict the air flow, causing moisture to build up on the seeds. If too much moisture builds up, the seeds may turn rotted or moldy. Tying a bag over the head prevents birds, squirrels, and most other wildlife from sneaking in and "harvesting" your sunflower seeds before you do. It also prevents the seeds from dropping to the ground and being lost. If the bag gets wet or torn, carefully remove it and replace it with a fresh, intact paper bag.  You may be able to prevent the bag from getting wet during a rainstorm by temporarily placing a plastic bag over it. Do not tie the plastic bag onto the flower head, though, and remove it as soon as the rain lets up to prevent mold from setting in. Change the paper bag as soon as it gets wet. A wet paper bag is more likely to be torn, and mold may even develop on the seeds if they sit in a wet bag for an extended period of time. Collect any seeds that may have dropped off into the old bag when changing it out. Examine the seeds for signs of possible damage, and if they are in good shape, store them in airtight containers until you are ready to harvest the remaining seeds. Once the back of the flower heads turn brown, cut them off and prepare to harvest the seeds.  Leave roughly 1 ft (30.5 cm) of stem attached to the flower head. Make sure that the paper bag is still securely tied onto the flower head. If it slips off as you remove and transport the sunflower head, you may lose a significant number of seeds.

Summary:
Wait until the sunflower begins to wilt. Tie a paper bag over the head. Change the bag as needed. Cut off the heads.