This paper introduces two new results on Kloosterman sums and their minimal polynomials. We characterise ternary Kloosterman sums modulo 27. We also prove a congruence concerning the minimal polynomial over ℚ of a Kloosterman sum. This paper also serves as a survey of our recent results on binary Kloosterman sums modulo 16, 32, 64 and 128 with Petr Lisonek.