Snake graphs are planar graphs that are obtained by iteratively stacking square boxes to the top or the right of the previous square box. They are naturally bipartite graphs, so it is natural to ask how to count its perfect matchings. In this talk, I will present a way to encode the snake graphs, and reveal a surprising connection of the perfect matchings of snake graphs to continued fractions. This result is due to Canakci and Rchiffler. If time permits, I will also discuss some open problems I am interested in, connected to areas like ergodic theory and limits of graphs, to collect ideas from the audience.