  Given a function $f$ on the positive half-line $\R_+$ and a sequence (finite
or infinite) of points $X=\{x_k\}_{k=1}^\omega$ in $\R^n$, we define and study
matrices $\kS_X(f)=\|f(|x_i-x_j|)\|_{i,j=1}^\omega$ called Schoenberg's
matrices. We are primarily interested in those matrices which generate bounded
and invertible linear operators $S_X(f)$ on $\ell^2(\N)$. We provide conditions
on $X$ and $f$ for the latter to hold. If $f$ is an $\ell^2$-positive definite
function, such conditions are given in terms of the Schoenberg measure
$\sigma(f)$. We also approach Schoenberg's matrices from the viewpoint of
harmonic analysis on $\R^n$, wherein the notion of the strong $X$-positive
definiteness plays a key role. In particular, we prove that \emph{each radial
$\ell^2$-positive definite function is strongly $X$-positive definite} whenever
$X$ is separated. We also implement a "grammization" procedure for certain
positive definite Schoenberg's matrices. This leads to Riesz--Fischer and Riesz
sequences (Riesz bases in their linear span) of the form
$\kF_X(f)=\{f(x-x_j)\}_{x_j\in X}$ for certain radial functions $f\in
L^2(\R^n)$. Examples of Schoenberg's operators with various spectral properties
are presented.
