In computing, a stateful firewall (any firewall that performs stateful packet inspection (SPI) or stateful inspection) is a firewall that keeps track of the state of network connections (such as TCP streams, UDP communication) traveling across it. The firewall is programmed to distinguish legitimate packets for different types of connections. Only packets matching a known active connection will be allowed by the firewall; others will be rejected.
Stateful inspection, also referred to as Dynamic Packet Filtering, is a security feature often included in business networks. Check Point Software introduced stateful inspection in the use of its FireWall-1 in 1994. 
Before the advent of stateful firewalls, a stateless firewall, a firewall that treats each network frame (or packet) in isolation, was normal. Such packet filters operate at the Network Layer (layer 3) and function more efficiently because they only look at the header part of a packet. A drawback of pure packet filters is that they are stateless; they have no memory of previous packets which makes them vulnerable to spoofing attacks. Such a firewall has no way of knowing if any given packet is part of an existing connection, is trying to establish a new connection, or is just a rogue packet. Modern firewalls are connection-aware (or state-aware), offering network administrators finer-grained control of network traffic.
The classic example of a network operation that may fail with a stateless firewall is the File Transfer Protocol (FTP). By design, such protocols need to be able to open connections to arbitrary high ports to function properly. Since a stateless firewall has no way of knowing that the packet destined to the protected network (to some host's destination port 4970, for example) is part of a legitimate FTP session, it will drop the packet. Stateful firewalls solve this problem by maintaining a table of open connections and intelligently associating new connection requests with existing legitimate connections.
Early attempts at producing firewalls operated at the Application Layer, which is the very top of the seven-layer OSI model. This method required exorbitant amounts of computing power and is rarely used in modern implementations.
A stateful firewall keeps track of the state of network connections (such as TCP streams or UDP communication) and is able to hold significant attributes of each connection in memory. These attributes are collectively known as the state of the connection, and may include such details as the IP addresses and ports involved in the connection and the sequence numbers of the packets traversing the connection. Stateful inspection monitors incoming and outgoing packets over time, as well as the state of the connection, and stores the data in dynamic state tables. This cumulative data is evaluated, so that filtering decisions would not only be based on administrator-defined rules, but also on context that has been built by previous connections as well as previous packets belonging to the same connection.
The most CPU intensive checking is performed at the time of setup of the connection. Entries are created only for TCP connections or UDP streams that satisfy a defined security policy. After that, all packets (for that session) are processed rapidly because it is simple and fast to determine whether it belongs to an existing, pre-screened session. Packets associated with these sessions are permitted to pass through the firewall. Sessions that do not match any policy are denied, as packets that do not match an existing table entry.
In order to prevent the state table from filling up, sessions will time out if no traffic has passed for a certain period. These stale connections are removed from the state table. Many applications therefore send keepalive messages periodically in order to stop a firewall from dropping the connection during periods of no user-activity, though some firewalls can be instructed to send these messages for applications.
The stateful firewall depends on the three-way handshake sometimes described as "SYN,SYN-ACK,ACK" (showing the order of SYNchronization bit and ACKnowledge bit use) of the TCP protocol when the protocol being used is TCP; when the protocol is UDP, the stateful firewall does not depend on anything related to TCP. When a client initiates a new connection, it sends a packet with the SYN bit set in the packet header. All packets with the SYN bit set are considered by the firewall as NEW connections. If the service which the client has requested is available on the server, the service will reply to the SYN packet with a packet in which both the SYN and the ACK bit are set. The client will then respond with a packet in which only the ACK bit is set, and the connection will enter the ESTABLISHED state. Such a firewall will pass all outgoing packets through but will only allow incoming packets if they are part of an ESTABLISHED connection, ensuring that hackers cannot start unsolicited connections with the protected machine.
Many stateful firewalls are able to track the state of flows in connectionless protocols. UDP hole punching is the technique associated with UDP. Such sessions usually get the ESTABLISHED state immediately after the first packet is seen by the firewall. Sessions in connectionless protocols can only end by time-out.
By keeping track of the connection state, stateful firewalls provide added efficiency in terms of packet inspection. This is because for existing connections the firewall need only check the state table, instead of checking the packet against the firewall's rule set, which can be extensive. Also, the concept of deep packet inspection is unrelated to stateful firewalls, because of its stateful feature, which checks incoming traffic against its state table first instead of jumping to the firewall's rule set. In this case if the state table is matched, then it doesn't need deep packet inspection.
Application-level filters 
Packet filtering alone is not regarded as providing enough protection. In order to effectively block peer-to-peer-related network traffic, what is needed is a firewall that does application filtering, which can be regarded as an extension to stateful packet inspection. Stateful packet inspection can determine what type of protocol is being sent over each port, but application-level filters look at what a protocol is being used for. For example, an application-level filter might be able to tell the difference between HTTP traffic used to access a Web page and HTTP traffic used for file sharing, whereas a firewall that is only performing packet filtering would treat all HTTP traffic equally.
Even though application layer firewalls are more secure than packet filtering, they are generally slower than stateful inspection. Application-layer firewalls are sometimes implemented using application proxies. Two TCP connections are established: one between the packet source and the firewall, another between the firewall and the packet destination. Application proxies intercept arriving packets on behalf of the destination, examine application payload, and then relay permitted packets to the destination. Suspicious data is dropped and the client and server never communicate directly with each other. Proxies necessarily involve more protocol stack overhead than inspecting packets at the network layer. Furthermore, because a unique proxy is required for each application, proxy firewalls can be less flexible and slower to upgrade than stateful inspection firewalls. Nevertheless, because application-level proxies are application-aware, the proxies can more easily handle complex protocols like H.323 or SIP, which are used for videoconferencing and VoIP (Voice over IP).
There is a risk that vulnerabilities in individual protocol decoders could allow an attacker to gain control over the firewall. This concern highlights the need to keep firewall software updated.
See also 
- United States Patent 5,606,668
- Check Point's Press Release “Check Point Introduces Revolutionary Internet Firewall Product Providing Full Internet Connectivity with Security; Wins 'BEST OF SHOW' Award at Networld+Interop ‘94”. 1994-05-06
- Review of Tomato firewall "...both L7-Filter and IPP2P are explicitly unmaintained. Given the steady stream of security updates for protocol dissectors in WireShark, your editor has a hard time believing that these other classifiers can be completely free of security issues."
- Hacker pierces hardware firewalls with web page