The "Caveman Regimen" is an approach to skin-care that avoids soaps and creams, and advocates washing the face with water only. (There are some variations of the regimen, such as only using an oil-based cleanser vs soap and water, or the full caveman which is doing absolutely nothing - no washing at all, except the water that hits your face in the shower.)
There are claims that soap, other skin-care/acne products and scrubbing can cause acne, and that if that is the case, changing to the Caveman Regimen can improve or cure it. (It may not help with eczema or acne caused by other triggers, such as PCOS.
There is an e-book recommending the Caveman Regimen, and a large number of personal (anecdotal) testimonials:
Personal blog: Caveman Skin Regimen: Update on being done dealing with acne
Not even using water for 60 days - and then saying moderation was the key: CAVEMAN BEAUTY: Can You Really Cure Acne by NOT Washing Your Face for 60 Days?
Testimonial of water only for one week: Medical Daily
Many testimonials: Acne.org
Use of olive-oil: Reddit
Does the Caveman Regimen reduce the incidence of acne?