Abstract A definition of Fuzzy Plane Projective Geometry (FPPG) is proposed with three models. The qualitative difference between a PPG and FPPG is emphasized through the existence of what has been called a vertical class of fuzzy points. A precise statement of Desargues' proposition in this fuzzy setting is formulated as an independent axiom. The weaker version of Fuzzy Small Desargues' proposition is stated and its independence from the basic axioms has been established. Finally, fuzzy collineations, which are bijections of the point set that map lines to lines and vertical classes to vertical classes, are introduced with examples drawn from the three models. Analogues of the basic theorems on collineation, centre and axis are proposed with a large number of examples. The paper is closed with two conjectures on the existence of lines and barriers of invariant points.