List comprehensions are used for creating lists from existing lists and other iterable structures. List comprehensions typically follow formatting similar to set-builder notation in discrete mathematics.

Consider the following illustrative Python3 example for generating a list of squares:

>>> sq = []
>>> for i in range(10):
...  sq.append(i**2)
... 
>>> sq
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

The same could be achieved in the following list comprehension¹⁾

>>> [i**2 for i in range(10)]
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

Note that this basic syntax is easily extended to multiple loops and variables:

>>> [(x**2, y**2) for x in range(3) for y in range(3)]
[(0, 0), (0, 1), (0, 4), (1, 0), (1, 1), (1, 4), (4, 0), (4, 1), (4, 4)]

Conditionals:

>>> [(x, y) for x in [1,2,3] for y in [3,1,4] if x != y]
[(1,3), (1,4), (2,3), (2,1), (2,4), (3,1), (3,4)]

Repetitions:

>>> [i for i in range(1, 5) for j in range(i)]
[1, 2, 2, 3, 3, 3, 4, 4, 4, 4]

Processing non-range() iterables:

>>> vec = [-4, -2, 0, 2, 4]
>>> [x*2 for x in vec]
[-8, -4, 0, 4, 8]

Flattening a double list:

>>> vec = [[1,2,3], [4,5,6], [7,8,9]]
>>> [num for elem in vec for num in elem]
[1, 2, 3, 4, 5, 6, 7, 8, 9]