Periodicity property of DFT in time and frequency domain
If the expression that defines the DFT is evaluated for all integers
  $k$ instead of just for $k = 0, \dots, N-1$ , then the resulting
  infinite sequence is a periodic extension of the DFT, periodic with
  period $N$.
  
  The periodicity can be shown directly from the definition:
  
  $$ X_{k+N} \stackrel{\mathrm{def}}{=} \ \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} (k+N) n} = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} k n} \underbrace{e^{-2 \pi i n}}_{1} = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} k n} = X_k.$$ 
  
  Similarly, it can be shown that the IDFT formula leads to a periodic
  extension.
  
  Source: Wikipedia.


While the periodic property of DFT isn't very widely utilized, it often causes aliasing problems.

Source: http://www.dspguide.com/ch10/3.htm