In this talk I will present some results contained in my PhD thesis. I will first introduce double covers of curves and graphs and then talk about Prym varieties and the Abel-Prym map, which is joint work with Yoav Len. I will then give an explicit and intrinsic way of constructing the tropical Prym variety that is continuous under degeneration. As a result we obtain the continuity of the tropical Prym-Torelli map. Additionally, I will give an overwiew of tropical Brill-Noether theory focusing on the tropical version of Martens' theorem solving a conjecture stated by Jensen and Len with a strict assumption on the degree of the divisor. This is joint work with Angelina Zheng.