Elementary concepts; characterizations of $\overline E^1$ and $S^1$ Locally connected spaces; fundamental properties of the euclidean $n$-sphere Peano spaces; characterizations of $S^2$ and the 2-manifolds Non-metric $LC$ spaces, with applications to subsets of the 2-sphere Basic algebraic topology Local connectedness and local co-connectedness Application of homology and cohomology theory to the theory of continua Generalized manifolds; dualities of the Poincar\'e and Alexander type Further properties of $n$-GMS; regular manifolds and generalized $n$-cells Submanifolds of a manifold; decomposition into cells $Lc^k$ subsets of an $n$-GM Accessibility and its applications endix. Some unsolved problems.