Tropical geometry is young field of mathematics that connects algebraic geometry and combinatorics. It considers "combinatorial shadows" of classical algebraic objects, which preserve information while being more susceptible to discrete methods. Tropical geometry has proven useful in such subjects as polynomial implicitization, scheduling problems, and phylogenetics. Of particular interesest in this work is the application of tropical geometry to study curves (and other varieties) over non-Archimedean fields, which can be tropicalized to tropical curves (and other tropical varieties). Chapter 1 presents background material on tropical geometry, and presents two perspectives on tropical curves: the embedded perspective, which treats them as balanced polyhedral complexes in Euclidean space, and the abstract perspective, which treats them as metric graphs. This chapter also presents the background on curves over non-Archimedean fields necessary for the rest of this work, including the moduli space of curves of a given genus and the Berkovich analytic space associated to a curve. Chapters 2 and 3 study tropical curves embedded in the plane. Chapter 2 deals with tropical plane curves that intersect non-transversely, and opens with a result on which configurations of points in such an intersection can be lifted to intersection points of classical curves. It then moves on to present a joint work with Matthew Baker, Yoav Len, Nathan Pflueger, and Qingchun Ren that builds up a theory of bitangents of smooth tropical plane quartic curves in parallel to the classical theory. Chapter 3 presents joint work with Sarah Brodsky, Michael Joswig, and Bernd Sturmfels, and is a study of which metric graphs arise as skeletons of smooth tropical plane curves. We begin by defining the moduli space of tropical plane curves, which is the tropical analog of Castryck and Voight's space of nondegenerate curves in [CV09]. The first main theorem is that our space is full-dimensional inside of the tropicalization of the corresponding classical space, a result proved using honeycomb curves. The chapter proceeds to a computational study of the moduli space of tropical plane curves, and explicitly computes the spaces for genus up to 5. The chapter closes with both theoretical and computational results on tropical hyperelliptic curves that can be embedded in the plane. Chapter 4 presents joint work with Qingchun Ren and is an algorithmic treatment of a special family of curves over a non-Archimedean field called Mumford curves. These are of particular interest in tropical geometry, as they are the curves whose tropicalizations can have genus-many cycles. We build up a family of algorithms, implemented in sage [S+13], for computing many objects associated to such a curve over the field of p-adic numbers, including its Jacobian, its Berkovich skeleton, and points in its canonical embedding. Chapter 5 is joint work with Ngoc Tran, and is a departure from studying tropical curves. It considers what it means for matrix multiplication to commute tropically, both in the context of tropical linear algebra and by considering the tropicalization of the classical commuting variety, whose points are pairs of commuting matrices. We give necessary and sufficient conditions for small matrices to commute, and illustrate three different tropical spaces, each of which has some claim to being "the" space of tropical commuting matrices.