This textbook was completed in 2011 and was published for the first time in 2013,by the American Mathematical Society, almost 10 years ago. Now Beijing World Publishing Corporation has decided to reprint it for mainland China. This is the perfect opportunity to correct the mistakes or misprints of the first print. I would like to believe that there are no mistakes or misprints left, but I fear that zero-fault is only an asymptotic notion. That being said, this revision has also been an opportunity to update a few things that were undecided or unfinished at that time.

For example: the question of whether any local induction was an immersion was settled. In fact, this question was solved by Henri Joris already in 1982 but I was unaware of it: the semicubic y2 = x3 =  is a submanifold for the subset diffeology while the induction t→(t2,t3) that describes it is not an immersion. Thus, there are indeed local inductions that are not immersions, and the case is closed. This example was the last brick to understand and situate the different subcategories relative to each other: induction/subduction, local-induction/local-subduction and immersion/submersion. This example was an opportunity to refinethe notion of submanifolds in a diffeological space by distinguishing between simply submanifolds, embedded submanifolds and smoothly embedded submanifolds. These last ones do not only put into play the D-topologies of the space and thesubspace but also the germs of diffeomorphisms of the subspace that extend to theambient space. This is described in (art. 4.4, Note 2), (art. 2.13, Definition 2)and (art. 2.14).