The authors define the class of ancient low-entropy flows. They consist of all ancient unit-regular, cyclic, integral Brakke flows in \(\mathbb{R}^3\) with entropy at most \(\sqrt{2\pi/e}.\) The main result is that these flows can be classified and it turns out that five types occur: a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, an ancient oval. Using this classification the mean convex neighborhood conjecture is considered.