We prove (see Theorem 1) that the set of maps (resp. surjective maps) with the shadowing property is C0C0-residual in the space of all continuous maps (resp. continuous surjective maps) of a compact topological manifold (possibly with boundary), which extends the similar result known for homeomorphisms (Pilyugin and Plamenevskaya, 1999) [22], as well as generalize the analogous result for maps (Kos̀cielniak et al., Preprint 2013) [12], obtained under an additional assumption that the manifold admits some kind of a piecewise linear structure. Moreover, we prove (see Theorem 2) that the set of maps (resp. surjective maps) with the s-limit shadowing property is C0C0-dense in the space of all continuous maps (resp. continuous surjective maps).