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You may be interested in reading these excellent [lecture notes](https://web.archive.org/web/20130127030523/https://math.mit.edu/~stevenj/18.303/separation.pdf) on the method of separation of variables by Steven Johnson. 

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**Quick Overview of Key Takeaways**

The notes highlight two main points:
- The method of separation of variables is applicable primarily in scenarios characterized by symmetry, such as time invariance, translational invariance, and rotational invariance. (For further insights on symmetry, see Derek's detailed answer.)
- Most analytically solvable partial differential equations (PDEs) are amenable to separation of variables.

**Categories of Applicability:**
1. **Separation in Time:** For time-invariant linear PDEs of the form $\partial_tu = Au$, where $A$ is a time-independent operator with a _complete_ basis, the solution can be expressed as:
   $$
   u(x,t) = \sum_n c_n(t) u_n(x)
   $$
   Section 2.1 of the notes elaborates on the conventional teaching of separation of variables, which generally involves identifying the eigenfunctions of the operator $A$.

2. **Separation in Space:** This method is typically used in cases with special spatial symmetries, such as in domains shaped like a box, a ball, an infinite tube, or combinations of these.