  We report on some recent work on deformation of spaces, notably deformation
of spheres, describing two classes of examples. The first class of examples
consists of noncommutative manifolds associated with the so called
$\theta$-deformations which were introduced out of a simple analysis in terms
of cycles in the $(b,B)$-complex of cyclic homology. These examples have
non-trivial global features and can be endowed with a structure of
noncommutative manifolds, in terms of a spectral triple $(\ca, \ch, D)$. In
particular, noncommutative spheres $S^{N}_{\theta}$ are isospectral
deformations of usual spherical geometries. For the corresponding spectral
triple $(\cinf(S^{N}_\theta), \ch, D)$, both the Hilbert space of spinors $\ch=
L^2(S^{N},\cs)$ and the Dirac operator $D$ are the usual ones on the
commutative $N$-dimensional sphere $S^{N}$ and only the algebra and its action
on $\ch$ are deformed. The second class of examples is made of the so called
quantum spheres $S^{N}_q$ which are homogeneous spaces of quantum orthogonal
and quantum unitary groups. For these spheres, there is a complete description
of $K$-theory, in terms of nontrivial self-adjoint idempotents (projections)
and unitaries, and of the $K$-homology, in term of nontrivial Fredholm modules,
as well as of the corresponding Chern characters in cyclic homology and
cohomology.
