These problems, diseases and pests are known to affect Chili pepper plants:
Blossom-end rot, Damping-off, Mosaic Virus, Bacterial spot, Southern blight.2
Blister beetles, Cutworms, Tomato hornworms, Pepper weevils, Root-knot nematodes.2
Tobacco Mosaic Virus : Tobacco Mosaic Virus (TMV) is carried by tobacco in cigarettes, and is mainly spread by infected hands touching plants or gardening equipment.
Aphids : Also known as greenfly and blackfly, Aphids are a common sap-sucking garden pest.
White Fly : Small, white winged insects measuring between 1-2 mm (1/16th")
Verticillium Wilt : Caused by soil-borne fungi that attacks through the roots and grows upward into the plant, causing wilting of the plant due to water stress. Affects many species and can survive periods of cold weather.
Thrips : Thrips are tiny white coloured flies that feed by burrowing into plant tissue - especially new buds and leaf growth.
Spider Mites : Tetranychus urticae (an animal with over 60 common names, including red spider mite and two-spotted spider mite) is one of many species of plant-feeding mites found in dry environments, and generally considered a pest. It is the most widely known member of the family Tetranychidae or Spider mites. [source http://en.wikibooks.org/wiki/Horticulture/Tetranychus_urticae]
Slugs : A snail without a shell. A mollusc with an elongated, soft body with a shiny appearance due to the slime coating.
Powdery Mildew : White powdery spot disease that affects a large variety of plants.
Blossom End Rot : A dark, indented, irregular-shaped patch that develops at the inferior aspect of the blossom or fruit, travelling out- and up-ward. Caused by an overall lack of calcium in the plant, which itself can be a symptom of multiple issues, including pH imbalance, over-fertilizing, under-watering, planting too soon (too cold). This is not a contagious 'disease' but rather a physiological disorder and lack of nutrients the fruit requires to produce healthy fruit. It cannot "spread" to adjacent plants, although if planted in similar conditions, other plants may suffer the same result.
Tobacco Etch Virus : Tobacco Etch Virus (TEV) is caused when infected aphids and other insects come into direct contact with the plant. Difficult to control once found - removal of plants is often the only remedy. Tabasco Chile plants are particularly susceptible to this disease. Can also affect perennial weed varieties, these then spread the disease to crops nearby.