A routing audit checks how well network routing is working by tracing the path packets take between hosts and determining if there are any unnecessary hops, loops, or multiple paths. The document discusses how to perform a routing audit using traceroute and analyzing TTL values if ICMP is blocked. Potential causes of routing issues are listed such as improperly configured routing protocols. The author recommends testing routing from different devices on the network and automating routing tests to perform periodically.