  A clear understanding of topology of higher-dimensional objects is important
in many branches of both pure and applied mathematics. In this survey we
attempt to present some results of higher-dimensional topology in a way which
makes clear the visual and intuitive part of the constructions and the
arguments. In particular, we show how abstract algebraic constructions appear
naturally in the study of geometric problems. Before giving a general
construction, we illustrate the main ideas in simple but important particular
cases, in which the essence is not veiled by technicalities.
  More specifically, we present several classical and modern results on the
embedding and knotting of manifolds in Euclidean space. We state many concrete
results (in particular, recent explicit classification of knotted tori). Their
statements (but not proofs!) are simple and accessible to non-specialists. We
outline a general approach to embeddings via the classical van
Kampen-Shapiro-Wu-Haefliger-Weber 'deleted product' obstruction. This approach
reduces the isotopy classification of embeddings to the homotopy classification
of equivariant maps, and so implies the above concrete results. We describe the
revival of interest in this beautiful branch of topology, by presenting new
results in this area (of Freedman, Krushkal, Teichner, Segal, Spiez and the
author): a generalization the Haefliger-Weber embedding theorem below the
metastable dimension range and examples showing that other analogues of this
theorem are false outside the metastable dimension range.
