Oil may cause stains on some material. Test on a corner first.
Sunlight will make this most effective. Dappled shade is a compromise that will reduce color fading.
Take a cloth made from microfiber or another lint-free material. Rinse out the cloth with very hot water, then wring it out thoroughly. Wrap a dry towel around the cloth and wring it out again. Remove the cloth, which should now be barely damp at all. Carefully wipe away dirt from the cover, and very lightly from the page edges. Buff with a dry cloth immediately afterward.
Don’t be tempted to use a hair dryer, oven, or fan. This can easily damage pages and loosen them from the spine.
If you see a furry white or grey mold on the pages, gently wipe it away with a soft cloth.