  We construct continuous families of Riemannian metrics on certain simply
connected manifolds with the property that the resulting Riemannian manifolds
are pairwise isospectral for the Laplace operator acting on functions. These
are the first examples of simply connected Riemannian manifolds without
boundary which are isospectral, but not isometric. For example, we construct
continuous isospectral families of metrics on the product of spheres S^4\times
S^3\times S^3. The metrics considered are not locally homogeneous. For a big
class of such families, the set of critical values of the scalar curvature
function changes during the deformation. Moreover, the manifolds are in general
not isospectral for the Laplace operator acting on 1-forms.
