Adobe brick houses are generally plastered with mud mixed with straw. In the arid SW U. S., the house is re-plastered as needed. Some are up to 400 years old, and some Indian pueblo constructions may be older.
Nowadays, the mud plaster is often replaced by stucco.

Roofs on adobe houses that have been kept up often have a very slight slant, and rainwater runs off through spouts. The roof is tarred; in the old days often the roof was up to a foot thick, of soil on wood support.

Construction has become expensive in cities like Santa Fe, New Mexico, because of labor costs and time required in aging (letting the adobe 'bricks' dry completely before using them in construction).
More modern adobe houses often rest on concrete basements.
Often the appearance is kept, on a modern house, by applying stucco in a thick, less angular layer. From a distance this looks like adobe.
We lived in one in which the stucco was applied over hollow tiles (shaped a lot like concrete blocks that are now common in this style of construction).