  In this paper we quantize superconformal $\sigma$-models defined by worldline
supermultiplets. Two types of superconformal mechanics, with and without a DFF
term, are considered. Without a DFF term (Calogero potential only) the
supersymmetry is unbroken. The models with a DFF term correspond to deformed
(if the Calogero potential is present) or undeformed oscillators. For these
(un)deformed oscillators the classical invariant superconformal algebra acts as
a spectrum-generating algebra of the quantum theory. Besides the $osp(1|2)$
examples, we explicitly quantize the superconformally-invariant worldine
$\sigma$-models defined by the ${\cal N}=4$ $(1,4,3)$ supermultiplet (with
$D(2,1;\alpha)$ invariance, for $\alpha\neq 0,-1$) and by the ${\cal N}=2$
$(2,2,0)$ supermultiplet (with two-dimensional target and $sl(2|1)$
invariance). The parameter $\alpha$ is the scaling dimension of the $(1,4,3)$
supermultiplet and, in the DFF case, has a direct interpretation as a vacuum
energy. In the DFF case, for the $sl(2|1)$ models, the scaling dimension
$\lambda$ is quantized (either $\lambda=\frac{1}{2}+{\mathbb Z}$ or
$\lambda={\mathbb Z}$). The ordinary two-dimensional oscillator is recovered,
after imposing a superselection restriction, from the $\lambda=-\frac{1}{2}$
model. In particular a single bosonic vacuum is selected. The spectrum of the
unrestricted two-dimensional theory is decomposed into an infinite set of
lowest weight representations of $sl(2|1)$. Extra fermionic raising operators,
not belonging to the original $sl(2|1)$ superalgebra, allow (for
$\lambda=\frac{1}{2}+{\mathbb Z}$) to construct the whole spectrum from the two
degenerate (one bosonic and one fermionic) vacua.
