The area around your bathtub contains many substrates which encourage mold and mildew -- which are essentially the same thing -- to grow. They include the tile grout in the tub surround, the drain and -- most troublesome of all -- the caulking around the tub. Mold is not only unsightly, it can cause allergic reactions and respiratory problems and create slipping hazards. Soap, vinegar, bleach and baking soda can all help control it.
Several varieties of mold and mildew can grow in the bathroom and bathtub, and Stachybotrys chartarum, usually known by the fearsome moniker "toxic black mold," is one of them. Unlike some bathroom molds, which need little more than moisture to survive, fortunately toxic black mold feeds on cellulose, and it's usually confined to the caulking and wood trim around the tub enclosure, if you see it at all. Other types of mold and mildew aren't as dangerous. But unless you're a mold expert, you can't tell the difference between harmful and benign species, so it's best to remove all of them.
The largest mold colony in the bathtub is probably inside the drain, and although you may not be able to see it, you can probably detect the musty odor. Start the mold cleanup by removing the strainer or stopper, either using a screwdriver or simply lifting it out, depending on the type, and wash the stopper with soap and water. To clean the drain, sprinkle in 1/2 cup of baking soda and follow this with a cup of white vinegar. The mixture creates a disinfecting and deodorizing foam volcano that will fill the drain pipe. Allow it to dissipate, then flush the drain with boiling water and replace the strainer.
Despite the popular belief that bleach is the best cleaner for mold, the U.S. Environmental Protection Agency recommends detergent and water. Scrub the bathtub, tile or fiberglass walls with an abrasive sponge to remove all the blackening. Vinegar is useful for deep cleaning fixture handles, especially when white mineral deposits are present, because it dissolves these while it disinfects. It can damage tile grout and lift caulking though, so for cleaning these, use either a paste of baking soda and water or hydrogen peroxide and toothbrush for scrubbing. In severe cases, leave the paste in place and cover it, and the caulking, with plastic wrap for several hours. You may not be able to remove the mold growing behind the caulking, and in that case, you'll have to remove the moldy caulking and replace it.
Mold grows in the bathroom because of moisture, and controlling moisture is the best strategy for preventing mold. If the bathroom has an exhaust fan, make sure it's in good working order and use it. If there isn't a fan, open the door and a window when no one is using the bathroom. It also helps to squeegee the walls and wipe the bathtub each time you take a bath or shower. Periodically spraying and wiping the walls and the bathtub with a weak 1-in-4 solution of vinegar and water kills spores that have landed there and prevents them from growing.