There are quite a lot open problems on exponential sums in general. Take a look at the following PDF http://web.maths.unsw.edu.au/~igorshparlinski/CharSumProjects.pdf
If your interest are gauss sums, there is some connection with combinatorics, Bourgain and others have many paper about exponential sums in multiplicative subgroups (which are Gauss sums).
Take a look at http://math.ucr.edu/~mcc/paper/122%20NewExp.pdf
Well there is an organic connection between Gauss sums and weights in irreducible cyclic codes. Classical 70 ish papers by Bob McEliece, Philippe Delsarte etc...