This volume consists of the write-ups of six sets of talks or "exposes" given in the seminar "Models of Peano Arithmetic" at Paris VII. The first two exposes give a detailed account of the new incompleteness theorem of Kirby-Paris and of the Paris -Harrington formula. The latter is a generalization of the finite form of Ramsey's Theorem which is not provable in (first-order) Peano Arithmetic. The second pair of exposes deal with the relation between the "new" incomplete­ ness theorem and the "old" one and with extensions of the Paris-Harrington result to transfinite progressions of theories. The third pair of exposes develop the Gaifman technology of types over models of Peano Arithmetic. An analysis is made of extensions of models and applied to ob­ tain the Abramson-Harrington theorem on uncountable models with short sequences of order indiscernables.