The given function is $f(x) = -x^2 + 10x - 27$.

This is a quadratic function, which is a polynomial. The domain of any polynomial function is all real numbers because polynomials are defined for all values of $x$.

Thus, the domain of the function $f(x) = -x^2 + 10x - 27$ is:

$\boxed{(-\infty, \infty)}$

This means the function is defined for every real value of $x$.

Would you like more details or have any questions?

Here are some related questions for further understanding:

1. What is the range of the function $f(x) = -x^2 + 10x - 27$?
2. How can we determine the vertex of the quadratic function?
3. Can you graph the function to visualize its behavior?
4. How would the domain change if the function were a square root or rational function?
5. How do we find the intercepts of the quadratic function?

Tip: For quadratic functions, the domain is always all real numbers unless restricted by a square root or division by zero.