With the fast development of wireless networks and devices, Mobile IP is expected to be used widely so that mobile users can access the Internet anywhere, anytime without interruption. However, some problems, such as firewall traversal and use of private IP addresses, restrict use of Mobile IP. The objective of this thesis is to design original schemes that can enable a mobile node at abroad to access its home network as well as the Internet securely and that can help Mobile IP to be used widely and commercially. Our solutions are secure, efficient, and scalable. They can be implemented and maintained easily. In this thesis, we mainly consider Mobile IPv4, instead of Mobile IPv6. Three research topics are discussed. In each topic, the challenges are investigated and the new solutions are presented. The first research topic solves the firewall traversal problems in Mobile IP. A mobile node cannot access its firewall-protected home network if it fails the authentication by the firewall. We propose that an IPsec tunnel be established between the firewall and the foreign agent for firewall traversal and that an IPsec transport security association be shared by the mobile node and a correspondent node for end-to-end security. The second topic researches further on firewall traversal problems and investigates the way of establishing security associations among network entities. A new security model and a new key distribution method are developed. With the help of the security model and keys, the firewall and the relevant network entities set up IPsec security associations to achieve firewall traversal. A mobile node from a private home network cannot communicate with other hosts with its private home address when it is visiting a public foreign network. A novel and useful solution is presented in the third research topic. We suggest that the mobile node use its Network Access Identifier (NAI) as its identification and obtain a public home address from its home agent. In addition, a new tunnel between the mobile node and its home agent is proposed.