Ivelisse Rubio University of Puerto Rico The divisibility of exponential sums has been used to characterize and prove properties in coding theory, cryptography and solvability of polynomial equations. In general, algebraic methods to estimate the p-divisibility of exponential sums over finite fields are non-elementary. The covering method provides an elementary and intuitive way to determine p-divisibility, which is particularly convenient in the applications. In this talk I will give an overview of the covering method for computing the p-divisibility of exponential sums and explain how it can be used in some applications. I will also mention other two problems and how I have re-visited them at different times during the last 30 years! BIO: Ivelisse Rubio has PhD in Applied Mathematics from Cornell University and is Professor at the University of Puerto Rico, Río Piedras. Her research is in the area of finite fields. She co-founded the REUs SIMU (1998-2002) and MSRI-UP (2007-2015). In 2006 SIMU received the American Mathematical Society's inaugural award to "Programs that make a difference. Ivelisse has received a SACNAS Presidential Service Award and the Dr. Etta Z. Falconer Award. She was a member of the Editorial Board of the American Mathematical Monthly, and is currently a member of the US National Committee for Mathematics and a member at-large of the Executive Committee of the Association for Women in Mathematics.