Abstract: Monolayers of graphite have attracted much experimental and theoretical attention in recent years owing to their unusual band structure and internal spinlike degrees of freedom. Graphene has also been suggested as a good candidate for spin-based quantum computing and spintronics, as it is expected to have long spin decoherence or relaxation times. In this talk, I will discuss spin and valley-spin dependent transport in graphene. First, I will focus on graphene nanoribbons and discuss how the zero energy modes localized at their edges get magnetized and can be used for spin injection. Furthermore, nanoribbons with rough edges exhibit mesoscopic spin conductance fluctuations with a universal value of $rms G_s \approx 0.4 e/4pi$, when the localization length is comparable to the length of the ribbon. Next, I will focus on graphene quantum dots and show how universalities featured by their spectra and conductance depend crucially on the edges of the dot. Finally, I will focus on a semiclassical theory of quantum transport in graphene nanostructures that we have developed recently and discuss how this edge dependence originates from the interplay of spin and valley-spin dynamics within the graphene quantum dot.