What problems remain unanswered today regarding the nature of variables? So far, to me, the greatest difficulty in studying philosophy is to recognize the importance of the problems: Exactly what make philosophers think these problems are worthy subject of study? Take Russell's The Principle of Mathematics §6 for example: 

The notion of the variable is one of the most difficult with which
  Logic has to deal, and in the present work a satisfactory theory as to
  its nature, in spite of much discussion, will hardly be found.

I wonder, as of today, what questions regarding variables remain unanswered. 
I'm aware of the prevailing prejudice in the area. I appreciate it if you don't automatically assume a dismissive tone.
 A: It seems to me that the modern approach to logic is "very far" from teh original view of W&R.
See Alfred North Whitehead & Bertrand Russell, Principia Mathematica, Introduction, Ch.I : PRELIMINARY EXPLANATIONS OF IDEAS AND NOTATIONS, page 4-on.
According to that view, what we today call "the connectives" are propositional functions; see page 6 :

The Logical Sum is a propositional function with two arguments $p$ and $q$,
  and is the proposition asserting $p$ or $q$ disjunctively, that is, asserting that at least one of the two $p$ and $q$ is true. This is denoted by $p \lor q$. Thus $p \lor q$ is the logical sum with $p$ and $q$ as arguments.

There is not the "modern" emphasis" on syntax : the initila list of symbols forming the alphabet, the definition of expression as a finite string of symbols, the recursive definition of formula as a specific type of expression, ...
Basically, W&R uses a "perfect" language where all the symbols denotes something : the symbols $\lor$ stays for the Logical Sum propositional function, and (presumibely) propositional functions are some sort of object in the world "out there" (recall Frege : the concept of function was basic and he "struggled" a lot with the issue of the denotation (Bedeutung) of such an "unsaturated" entity ...).
If so, for what kind of object the "variable symbols" stand for ?
See page 4 :

To sum up, the three salient facts connected with the use of the variable
  are: (1) that a variable is ambiguous in its denotation and accordingly undefined [...].

In a modern logic textbook we simply have symbols and interpretations, and some cunning device to assign a "temporary" denotation to variables in order to determine the meaning (an truth-value) of an expression with a variable inside.
Thus, a variable is like a pronoun of naural language; in "It is red", the pronoun does not denote outside the context where the sentence is uttered. If I'm uttering it now, it denotes the red book on my desk.
The device of "variable assignment" used by math logic in the recursive semantical clauses for a predicate logic language has exactly the same function : to give denotation to a variable in the context of an interpretation.

In conclusion : so what ? Have we solved the problem or only skipped it ?
We can consider the influence of Wittgenstein : he was absolutely crucial, with its move from the "perfect language" considered into the Tractatus to its second phase regarding "linguistic games" and so on, for leaving the idea of a language where every part of it must denote somethig ... 
