Byzantine agreement is a fundamental problem in fault-tolerant distributed networks that has been studied intensively for the last four decades. Most of these works designed protocols for complete networks. A key goal in Byzantine protocols is to tolerate as many Byzantine nodes as possible. The work of Dwork, Peleg, Pippenger, and Upfal [STOC 1986, SICOMP 1988] was the first to address the Byzantine agreement problem in sparse, bounded degree networks and presented a protocol that achieved almost-everywhere agreement among honest nodes. In such networks, all known Byzantine agreement protocols (e.g., Dwork, Peleg, Pippenger, and Upfal, STOC 1986; Upfal, PODC 1992; King, Saia, Sanwalani, and Vee, FOCS 2006) that tolerated a large number of Byzantine nodes had a major drawback that they were not fully-distributed -- in those protocols, nodes are required to have initial knowledge of the entire network topology. This drawback makes such protocols inapplicable to real-world communication networks such as peer-to-peer (P2P) networks, which are typically sparse and bounded degree and where nodes initially have only local knowledge of themselves and their neighbors. Indeed, a fundamental open question raised by the above works is whether one can design Byzantine protocols that tolerate a large number of Byzantine nodes in sparse networks that work with only local knowledge, i.e., fully-distributed protocols. The work of Augustine, Pandurangan, and Robinson [PODC 2013] presented the first fully-distributed Byzantine agreement protocol that works in sparse networks, but it tolerated only up to $O(\sqrt{n}/ polylog(n))$ Byzantine nodes (where $n$ is the total network size). We answer the earlier open question by presenting fully-distributed Byzantine agreement protocols for sparse, bounded degree networks that tolerate significantly more Byzantine nodes -- up to $O(n/ polylog(n))$ of them.