In the beginning of this very interesting article (which is the first part of an overview of topological descent theory) an elementary topological approach to Grothendieck's idea of descent is given. The authors work also with the subcategories \({\mathcal E}(B)\) of bundles over \(B\) given by a suitable class \({\mathcal E}\) of continuous maps (i.e., with a subfibration of the basic fibration given by \(Top/B\)), so that when \({\mathcal E}\) is the class of local homeomorphisms, \({\mathcal E}(B)\) is actually the category of set-valued sheaves on \(B\). In the second section it is shown how the theory of monads provides a direct categorical approach to descent theory. The monadic description covers descent also in the abstract context of a bifibred category satisfying the Beck-Chevalley condition. The number of user-friendly descent criteria, in particular in the realm of exact categories and of locally cartesian closed categories are also presented. The last section of this paper is devoted to the investigation of (effective) \({\mathcal E}\)-descent maps of \(Top\) when \({\mathcal E}\) is the class of all continuous maps (``global-descent''), of open-subspace embeddings (``open-descent''), or of local homeomorphisms (``étale-descent''). A complete characterization of étale-descent maps is given and it is shown that effective global-descent maps are effective étale-descent maps. The examples demonstrate the subtlety of the descent problem in concrete situations.