Do the digits of $\pi$ contain every possible finite-length digit sequence?

Question

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Prove there are no hidden messages in Pi

_{This is not a practical problem. I am asking out of curiosity. Any links/references are most welcome.}

Say, we write the digits of $\pi$ in base $10$. Does this sequence of digits contain every possible finite length digit sequence? What about $e$, $\sqrt{2}$ or some other commonly known irrational numbers?

Is this property of numbers independent of base? If a number has this property when written in base $10$, will it also have it in base $2$, $3$ and all other bases?

Answer 1

According to the Wikipedia article on normal numbers, "It is not even known whether all digits occur infinitely often in the decimal expansions of those constants."