  These notes are an expanded version of two series of lectures given at the
winter school in mathematical physics at les Houches and at the Vietnamese
Institute for Mathematical Sciences. They are an introduction to factorization
algebras, factorization homology and some of their applications, notably for
studying $E_n$-algebras. We give an account of homology theory for manifolds
(and spaces), which give invariant of manifolds but also invariant of
$E_n$-algebras. We particularly emphasize the point of view of factorization
algebras (a structure originating from quantum field theory) which plays, with
respect to homology theory for manifolds, the role of sheaves with respect to
singular cohomology. We mention some applications to the study of mapping
spaces and study several examples, including some over stratified spaces.
