  Let $G$ be a compact group. For $1\leq p\leq\infty$ we introduce a class of
Banach function algebras $\mathrm{A}^p(G)$ on $G$ which are the Fourier
algebras in the case $p=1$, and for $p=2$ are certain algebras discovered in
\cite{forrestss1}. In the case $p\not=2$ we find that $\mathrm{A}^p(G)\cong
\mathrm{A}^p(H)$ if and only if $G$ and $H$ are isomorphic compact groups.
These algebras admit natural operator space structures, and also weighted
versions, which we call $p$-Beurling-Fourier algebras. We study various
amenability and operator amenability properties, Arens regularity and
representability as operator algebras. For a connected Lie $G$ and $p>1$, our
techniques of estimation of when certain $p$-Beurling-Fourier algebras are
operator algebras rely more on the fine structure of $G$, than in the case
$p=1$. We also study restrictions to subgroups. In the case that $G=SU(2)$,
restrict to a torus and obtain some exotic algebras of Laurent series. We study
amenability properties of these new algebras, as well.
