A Barker sequence is a finite length binary sequence with the minimum possible aperiodic autocorrelation. Currently, only eight known Barker sequences exist and it has been conjectured that these are the only Barker sequences that exist. This thesis proves that long sequences (having length longer than thirteen) must have an even length and be a perfect square. Barker sequences are then used to explore flatness problems related to Littlewood polynomials. These theorems could be used to determine the existence or non-existence of longer sequences. Lastly, an application of Barker sequences is given. Barker sequences were initially investigated for the purposes of pulse compression in radar systems. This technique results in better range and Doppler resolution without the need to shorten a radar pulse, nor increase the power.