Answer:

1.      Each number is the sum of the two numbers before it. 

For example:         1 + 1 = 2, then 1 + 2 = 3, then 2 + 3 = 5, etc.

2.      The next three numbers in the sequence are 13 (5 + 8), 21 (8 + 13), and 34 (13 + 21).

      This is known as the Fibbonaci sequence.

Note:  Fibonacci (born 1170) was an Italian mathematician.   It was in the Liber Abaci that the problem which leads to his famous sequence was posed and solved:

How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?

The solution gives rise to the sequence beginning 1, 1, ..., and in which each succeeding term is given by adding the previous two.
Cf. Fibonacci sequence.