  We exhibit large classes of examples of noncommutative finite-dimensional
manifolds which are (non-formal) deformations of classical manifolds. The main
result of this paper is a complete description of noncommutative
three-dimensional spherical manifolds, a noncommutative version of the sphere
$S^3$ defined by basic K-theoretic equations. We find a 3-parameter family of
deformations $S^3_{\mathbf{u}}$ of the standard 3-sphere $S^3$ and a
corresponding 3-parameter deformation of the 4-dimensional Euclidean space
$\mathbb R^4$. For generic values of the deformation parameters we show that
the obtained algebras of polynomials on the deformed $\mathbb R^4_{\mathbf{u}}$
only depend on two parameters and are isomorphic to the algebras introduced by
Sklyanin in connection with the Yang-Baxter equation. It follows that different
$S^3_{\mathbf{u}}$ can span the same $\mathbb R^4_{\mathbf{u}}$. This
equivalence generates a foliation of the parameter space $\Sigma $. This
foliation admits singular leaves reduced to a point. These critical points are
either isolated or fall in two 1-parameter families $ C_{\pm}\subset \Sigma $.
Up to the simple operation of taking the fixed algebra by an involution, these
two families are identical and we concentrate here on $ C_{+}$. For $u \in
C_{+}$ the above isomorphism with the Sklyanin algebra breaks down and the
corresponding algebras are special cases of $\theta$-deformations, a notion
which we generalize in any dimension and various contexts, and study in some
details. Here, and this point is crucial, the dimension is not an artifact,
i.e. the dimension of the classical model, but is the Hochschild dimension of
the corresponding algebra which remains constant during the deformation....
