The video game based on the movie was released in 1990 and was available for the ZX Spectrum, Amiga, Atari ST, Atari 8-bit, IBM PC, Amstrad, Nintendo Entertainment System, Super NES and Game Boy.
The game featured deep sea combat, side-scrolling action, and cinematic sequences. The object is to evade destruction and eliminate saboteurs. The caterpillar drive is particularly useful for quietly escaping the enemy.
The Super NES version is one of only a few games that uses the Super Scope accessory, though it is only used to play bonus stages that put the submarine in first person view, where the player has to destroy a number of enemies and projectiles. The use of the Super Scope is optional.
The NES version has an exclusive level: the final stage changes to platform-style gameplay. The player, as Ramius, must find and disable bombs soviet-loyal crewmembers have set in the weapons bay of the Red October.
Source: Wikipedia, "The Hunt for Red October (video game) available under the CC-BY-SA License.