Convexity in graphs has been widely discussed in graph theory and G. Chartrand et al. introduced the convexity number of oriented graphs in 2002. In this thesis, we follow the notions addressed by them and develop an extension in some related topics of convexity in directed graphs. Let D be a connected oriented graph. A set S subseteq V(D) is convex in D if, for every pair of vertices x, yin S, the vertex set of every x-y geodesic (x-y shortest directed path) and y-x geodesic in D is contained in S. The convexity number con(D) of a nontrivial oriented graph D is the maximum cardinality of a proper convex set of D. We show that for every possible triple n, m, k of integers except for k=4, there exists a strongly connected digraph D of order n, size m, and con(D)=k. Let G be a graph. We define the convexity spectrum S_{C}(G)={con(D): D is an orientation of G} and the strong convexity spectrum S_{SC}(G)={con(D): D is a strongly connected orientation of G}. Then S_{SC}(G) ⊆ S_{C}(G). We show that for any n ¡Ú 4, 1 ≤ a ≤ n-2 and a n ¡Ú 2, there exists a 2-connected graph G with n vertices such that S_C(G)=S_{SC}(G)={a,n-1}, and there is no connected graph G of order n ≥ 3 with S_{SC}(G)={n-1}. We also characterizes the convexity spectrum and the strong convexity spectrum of complete graphs, complete bipartite graphs, and wheel graphs. Those convexity spectra are of different kinds. Let the difference of convexity spectra D_{CS}(G)=S_{C}(G)- S_{SC}(G) and the difference number of convexity spectra dcs(G) be the cardinality of D_{CS}(G) for a graph G. We show that 0 ≤ dcs(G) ≤ ⌊|V(G)|/2⌋, dcs(K_{r,s})=⌊(r+s)/2⌋-2 for 4 ≤ r ≤ s, and dcs(W_{1,n-1})= 0 for n ≥ 5. The convexity spectrum ratio of a sequence of simple graphs G_n of order n is r_C(G_n)= liminflimits_{n to infty} frac{|S_{C}(G_n)|}{n-1}. We show that for every even integer t, there exists a sequence of graphs G_n such that r_C(G_n)=1/t and a formula for r_C(G) in subdivisions of G.