This paper proposes two topology discovery algorithms for IP networks, namely, a network layer topology discovery algorithm and a link layer topology discovery algorithm. The network layer topology discovery algorithm discovers the subnets and devices in the network of interest and the connections among them. The devices in a subnet can be found by a network layer topology discovery algorithm; however, the connections among the devices cannot be obtained. The link layer topology discovery algorithm is proposed to find the devices in a subnet and the connections among them. The two algorithm are integrated to find the detailed topology map of an IP network. The proposed topology discovery algorithms are implemented based on the Tcl/Tk and Scotty environment. Some implementation details are discussed.