Languages of cardinality higher than $\aleph_0$
In traditional Formal Language theory, all languages are countable. This is because they're defined as a subset of $\Sigma^*$, so they always contain strings of finite length, meaning we can map them to integers.

We rarely deal with finite languages, since all finite languages are regular, and therefore trivially decidable.

There has been research into the computability of functions on real numbers. The difficulty here is that in practice, there's no single way to represent arbitrary real numbers. A common method is looking at, given a real-valued function $f$, inputs $\vec{x} $ and an integer $n$,
can we use a Turing Machine to compute a representation $f(\vec{x})$ to $n$ digits of precision?

You could view a real number as justing being a function taking $0$ arguments, so a real number $r$ is can be seen as a function from $n$ to the first $n$ binary digits of $r$.

Interestingly, since Turing Machines are finite, the set of all Turing Machines is countable, so the set of all computable real numbers is countable, meaning there are many more uncomputable reals than computable ones.

For more info, see http://en.wikipedia.org/wiki/Computable_number.

Note that there's also a concept of automata on infinite words, used in formal language theory and program verification. These languages can be uncountable: an infinite sequence of $0$s and $1$s is equivalent to a subset of $\mathbb{N}$ so an automaton that accepts all such words accepts an uncountable language.