We describe analogs of AF C*-algebras which act on Lp spaces.
the ``good'' ones (which we call spatial), there is an analog of the
Elliott classification theorem in the C* case. The spatial Lp
AF algebras also have good properties. The development requires
consideration of direct limit of Lp operator algebras and
unitization of Lp operator algebras.
This is joint work with Maria Grazia Viola.