U.S. Air & Surface-Launched Rockets
Air- Launched
The concept of rocket propulsion was not new in World War II, but it was the first time that extensive use of the propulsion system was made. First to mind probably come the German V-1 and V-2 rockets, and the rocket-driven fighters, but the Allies too employed this engine, if less spectecular.
    The original desire for rockets as weapons came from the British, who needed a more effective weapon than the depth charges. A rocket, because it was forward-firing, could serve to surprise a U-boat before it could dive. Like an archer's weapon of old, these first airborne rockets carried a simple steel head. If they hit a submarine,  they would penetrate    the steel hull  and hopefully prevent its diving. This weapon was 3.5" in diameter, and a explosive warhead could be fitted for use against all kinds of targets, though the power of that warhead was insufficent to damage major warships. This was among the reasons that a large rocket was developed using the 3.5" rocket motor in conjunction with a 5" AA shell. Much of the velocity inherent in the lighter rocket was lost, however. 
Stats 3.5" Rocket:  
Weight: 25kg / 55lbs  
Length: 1397mm / 55ins  
Warhead: 9kg / 20lbs solid steel or explosives  
Speed: 358m/s / 1175f/s
    The family of rockets based on the 3.5" rocket engine were initially launched by means of a rail extending along th eentire length of the rocket and attached to the wings of a plane. From this rail, the rocket was fired with great accuracy. It did, hoever, create an inordinate amount of drag, causing the loss of 17 knots of speed, an inacceptable condition for fighter planes, which were among the foremost platforms for the use of the new weapons. Testing finally resolved the issue in favor of a new method of firing, two hardpoints which provided sufficent stability to achieve reasonable accuracy at the usual ranges, at much reduced drag. 

    The first use of these rockets, as mentioned, was in ASW, and a quite successful start it was too. However, it was quickly established that this was not the only possible use of such weapons, which turned out correct -- it was a very accurate weapon for ground and naval targets alike. This usefulness triggered the development of another rocket that was designed with just these targets in mind. 

    The new rocket sported a 5" rocket engine, to regain the velocity lost with the adaption of a five inch shell to a 3.5" engine; in the end it turned out even fast than the regular 3.5" rocket, which resulted in its designation as High-Velocity Aircraft Rocket, or HVAR. The Pacific theater was only minor in regard to this rocket, but the earlier 5" variant was highly useful in attacking pillboxes, ships, and other point targets. Especially for the Pacific, however, the Navy  

undertook the development  of a weapon of incredible power, the so-called "Tiny Tim" 11.75"rocket. This awesome weapon was developed as a weapon useable by fighters against ships, especially such ships that were heavily protected. Against them, high-speed attacks were favorable, but torpedo-bombing and dive-bombing weren't fast enough. A rocket promised a quick run in and a weapons release 2000 yds off the target. 

    The strange diameter was dictated by the warhead -- a 227kg (500lbs) semi-armor piercing bomb -- and the casing of the rocket engine, a standard oil-well casing. It was not particularly accurate, with a 4000yds error, due to the fact that it could not be launched from rails or hardpoints, but had to be free-falling before the rocket engine couldignite. They were issued to carriers Franklin and Intrepid, but Franklin's damage in March 1945 prevented the carrier from using the new weapon, although they did serve to increase the carnage on the ship as they exploded underneath

Stats HVAR:  
Diameter: 5"  
Weight: 63.5kg / 140lbs  
Length: 1829mm / 72ins  
Warhead: 25kg / 55lbs  
Speed: 419m/s / 1375f/s
Stats "Tiny Tim":  
Diameter: 11.75"  
Weight: 581kg / 1280lbs  
Length: 3124mm / 123ins  
Warhead: 227kg / 500lbs  
Speed: 247m/s / 810f/s
armed fighters. Intrepid's results with the weapon are unknown. 
 Seaborne Systems
    The primary reason for the employment of rocket systems in seaborne invasions was a lack of intermediate support for the landing troops. The odd amphibious operation would be supported heavily by air strikes and ship artillery fire, which, however, had one critical flaw: it could not be timed too accurately. The possibility existed that, given the time between the lifting of both air- and sea bombardment, and the arrival of the troops ashore, enemy forces ashore previously pinned by the onslaught, would recover. Rockets, lifted close to shore in craft accompanying the landing craft, could cover the distance in short time and thus, a covering fire could be sustained until the very last moments before invasion.
    The first momentum for the development of such devices was gained when in mid-June 1942, Commander, Amphibious Forces, Pacific, Vice-Admiral Wilson Brown, attended a demonstration given for Mousetrap and other rocket systems. He suggested such rockets be used in support of amphibious landings to overcome the problems detailed above. A mere month later, using heavily parts already existing, CalTech, the developer of the U.S. Hedgehog, tested the first 4.5" rocket, using a Mousetrap rocket and a 20lbs bomb as head.
Data for Shore-Bombardment Rockets: 
Though available, it is not significantly different from that of air-launched weapons of the same diameters. Range is quoted in the text.
    First use of the new rocket system was made during the invasion of North Africa, when LCS, mounting two launchers with 12 rockets each, used it to shell Casablanca. 
The 5" aircraft rocket was used for a short while in lieu of more specialized rockets, but reload was laborious and led to the weapon's abandonment. 

    Since these weapons were short ranged, they forced upon the firing craft a undue closeness to the target, which might dislike being shot at and return fire; the benefits of the rocket as a weapon, however, precluded its abandonment. Ergo, a 10.000 yards system was developed, using both 3.5" and 5" rockets. These weapons, contrary to the 4.5" types, would be spin-stabilized, favored over the fin-stabilization because it would yield more rapid fire and shorter rockets. 

    It turned out that the 3.5" weapon did not provide sufficent accuracy at 10.000 yards, causing it to be not pursuit further. Rockets were one of the mainstays of amphibious support, being in every operation after "Dragoon", the landing in southern France. For the ships that carried these weapons, see the forthcoming section on amphibious craft.