We study the network structure and resilience of the sterling investment-grade and high-yield corporate bond markets. Using proprietary, transaction-level data, first we analyse the key properties of the trading networks in these markets. We find that the trading networks exhibit a core-periphery structure where a large number of non-dealers trade with a small number of dealers. Consistent with dealer behaviour in the primary market, we find that trading activity is particularly concentrated for newly issued bonds, where the top three dealers account for 45% of trading volume. Second, we test the resilience of these markets to the failure or paralysis of a key dealer, or to bond rating downgrades. We find that whilst the network structure has been broadly stable and the market broadly resilient around bond downgrades over our 2012-2017 sample period, the reliance on a small number of participants makes the trading network somewhat fragile to the withdrawal of a few key dealers from the market.