Write down the given information for the arithmetic progression.

The formula for the $n^{th}$ term of an arithmetic progression is $T_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.
We are given:
The $10^{th}$ term, $T_{10} = -17$.
The $18^{th}$ term, $T_{18} = -49$.

Using the formula, we can set up two equations:
For $T_{10}$:

$$a + (10-1)d = -17$$ $$a + 9d = -17 \quad (*)$$

For $T_{18}$:

$$a + (18-1)d = -49$$ $$a + 17d = -49 \quad (**)$$

Subtract equation $(*)$ from equation $(**)$:

$$(a + 17d) - (a + 9d) = -49 - (-17)$$ $$8d = -49 + 17$$ $$8d = -32$$ $$d = \frac{-32}{8}$$ $$d = -4$$

Substitute $d = -4$ into equation $(*)$:

$$a + 9(-4) = -17$$ $$a - 36 = -17$$ $$a = -17 + 36$$ $$a = 19$$

Now that we have $a = 19$ and $d = -4$, we can find $T_{27}$:

$$T_{27} = a + (27-1)d$$ $$T_{27} = a + 26d$$ $$T_{27} = 19 + 26(-4)$$ $$T_{27} = 19 - 104$$ $$T_{27} = -85$$

The $27^{th}$ term of the arithmetic progression is $\boxed{-85}$.