It is popularly suggested that cockroaches will "inherit the earth".
The four main species are the American, German, Asian, and Oriental cockroaches. They vary in size from about 1/2" (German), up to 2" (American). In America, the species that is most common in homes would most likely be the German variety.

Cockroaches increase in population at a very rapid rate. The German cockroach female lives about 200 days. Each egg case that she lays includes thirty to fifty eggs. In her lifetime, she will produce between four to six egg cases. German cockroaches are clearly able to thrive
Cockroaches are one of the most commonly noted household pest insects. They feed on human and pet food, and can leave an offensive odor.They can also passively transport microbes on their body surfaces including those that are potentially dangerous to humans.
•           A cockroach can live almost a month without food.
•           A cockroach can live about two weeks without water.

Cockroaches crawl through dirty areas and then walk around our homes tracking in lots of bacteria and germs. They can contaminate food by shedding their skins. Their cast off skin and waste byproducts are allergens that can trigger allergic reactions, asthma and other illnesses, especially in children.
If you spot a cockroach at home or work, there are likely more of them concealed nearby. Contact a pest professional to avoid letting an infestation get out of hand.