  We introduce and study series expansions of real numbers with an arbitrary
Cantor real base $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$, which we call
$\boldsymbol{\beta}$-representations. In doing so, we generalize both
representations of real numbers in real bases and through Cantor series. We
show fundamental properties of $\boldsymbol{\beta}$-representations, each of
which extends existing results on representations in a real base. In
particular, we prove a generalization of Parry's theorem characterizing
sequences of nonnegative integers that are the greedy
$\boldsymbol{\beta}$-representations of some real number in the interval
$[0,1)$. We pay special attention to periodic Cantor real bases, which we call
alternate bases. In this case, we show that the $\boldsymbol{\beta}$-shift is
sofic if and only if all quasi-greedy $\boldsymbol{\beta}^{(i)}$-expansions of
$1$ are ultimately periodic, where $\boldsymbol{\beta}^{(i)}$ is the $i$-th
shift of the Cantor real base $\boldsymbol{\beta}$.
