  A Buchdahl star is a highly compact star for which the boundary radius $R$
obeys $R=\frac98 r_+$, where $r_+$ is the gravitational radius of the star
itself. A quasiblack hole is a maximum compact star, or more generically a
maximum compact object, for which the boundary radius $R$ obeys $R=r_+$.
Quasiblack holes are objects on the verge of becoming black holes. Continued
gravitational collapse ends in black holes and has to be handled with the
Oppenheimer-Snyder formalism. Quasistatic contraction ends in a quasiblack hole
and should be treated with appropriate techniques. Quasiblack holes, not black
holes, are the real descendants of Mitchell and Laplace dark stars. Quasiblack
holes have many interesting properties. We develop the concept of a quasiblack
hole, give several examples of such an object, define what it is, draw its
Carter-Penrose diagram, study its pressure properties, obtain its mass formula,
derive the entropy of a nonextremal quasiblack hole, and through an extremal
quasiblack hole give a solution to the puzzling entropy of extremal black
holes.
