What are the most attractive Turing undecidable problems in mathematics? 

There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on the Wikipedia page.

Standard community wiki rules. One example per post please. I will accept the answer I find to be the most attractive, according to the following criteria:


Examples must be undecidable in the sense of Turing computability. (Please not that this is not the same as the sense of logical independence; think of word problem, not Continuum Hypothesis.)
The best examples will arise from natural mathematical questions.
The best examples will be easy to describe, and understandable by most or all mathematicians. 
(Challenge) The very best examples, if any, will in addition have intermediate Turing degree, strictly below the halting problem. That is, they will be undecidable, but not because the halting problem reduces to them. 


Edit: This question is a version of a previous question by Qiaochu Yuan, inquiring which problems in mathematics are able to simulate Turing machines, with the example of the MRDP theorem on diophantine equations, as well as the simulation of Turing machines via PDEs. He has now graciously merged his question here.

The mortality problem for $3\times 3$ matrices: given a finite set $F$ of $3\times 3$
integer matrices, decide whether the zero matrix is a product of members of $F$ 
(with repetitions allowed). This was proved unsolvable by Michael Paterson, 
Studies in Applied Mathematics 49 (1970), 105--107, doi: 10.1002/sapm1970491105.

The corresponding problem for $2\times 2$ matrices is apparently still open.

Edit (11 September 2016): The problem for $2\times 2$ matrices is apparently still open, despite the solution of a seemingly similar problem mentioned in Igor Potapov's answer below.
