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Not especially famous, long-open problems which anyone can understand

Which journals publish expository work?

Is there an introduction to probability theory from a structuralist/categorical perspective?

Analytic tools in algebraic geometry

Experimental mathematics leading to major advances

Homotopy groups of Lie groups

Why is the Hodge Conjecture so important?

Where does a math person go to learn statistical mechanics?

What is the definition of "canonical"?

What's the "best" proof of quadratic reciprocity?

What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

Why do Bernoulli numbers arise everywhere?

What is entropy, really?

How do I see LaTeX math on any web page and in email?

Why should I believe the Mordell Conjecture?

What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Algebraic geometry examples

(Preferably rare) Audio/Video recordings of famous mathematicians?

An example of a beautiful proof that would be accessible at the high school level?

What is Lagrange Inversion good for?

Relating Category Theory to Programming Language Theory

Is a "non-analytic" proof of Dirichlet's theorem on primes known or possible?

Galoisian sets of prime numbers

Open project: Let's compute the Fourier expansion of a non-solvable algebraic Maass form.

Shortest/Most elegant proof for $L(1,\chi)\neq 0$

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Is there a "Basic Number Theory" for elliptic curves?

Hirzebruch's motivation of the Todd class

Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications

Euler-Maclaurin formula and Riemann-Roch