Artillery
Bombards developed in Europe were massive smoothbore weapons distinguished by their lack of a field carriage, immobility once emplaced, highly individual design, and noted unreliability (in 1460 James II, King of Scots, was killed when one exploded at the siege of Roxburgh). Their large size precluded the barrels being cast and they were constructed out of metal staves or rods bound together with hoops like a barrel, giving their name to the gun barrel.
The use of the word "cannon" marks the introduction in the 15th century of a dedicated field carriage with axle, trail and animal-drawn limber—this produced mobile field pieces that could move and support an army in action, rather than being found only in the siege and static defenses. The reduction in the size of the barrel was due to improvements in both iron technology and gunpowder manufacture, while the development of trunnions—projections at the side of the cannon as an integral part of the cast—allowed the barrel to be fixed to a more movable base, and also made raising or lowering the barrel much easier.

The first land-based mobile weapon is usually credited to Jan Žižka, who deployed his oxen-hauled cannon during the Hussite Wars of Bohemia (1418–1424). However, cannons were still large and cumbersome. With the rise of musketry in the 16th century, cannon were largely (though not entirely) displaced from the battlefield—the cannon were too slow and cumbersome to be used and too easily lost to a rapid enemy advance.
The combining of shot and powder into a single unit, a cartridge, occurred in the 1620s with a simple fabric bag, and was quickly adopted by all nations. It speeded loading and made it safer, but unexpelled bag fragments were an additional fouling in the gun barrel and a new tool—a worm—was introduced to remove them. Gustavus Adolphus is identified as the general who made cannon an effective force on the battlefield—pushing the development of much lighter and smaller weapons and deploying them in far greater numbers than previously.  The outcome of battles was still determined by the clash of infantry.
Shells, explosive-filled fused projectiles, were in use by the 15th century. The development of specialized pieces—shipboard artillery, howitzers and mortars—was also begun in this period. More esoteric designs, like the multi-barrel ribauldequin (known as "organ guns"), were also produced.
The 1650 book by Kazimierz Siemienowicz Artis Magnae Artilleriae pars prima was one of the most important contemporary publications on the subject of artillery. For over two centuries this work was used in Europe as a basic artillery manual.
One of the most significant effects of artillery during this period was however somewhat more indirect—by easily reducing to rubble any medieval-type fortification or city wall (some which had stood since Roman times), it abolished millennia of siege-warfare strategies and styles of fortification building. This led, among other things, to a frenzy of new bastion-style fortifications to be built all over Europe and in its colonies, but also had a strong integrating effect on emerging nation-states, as kings were able to use their newfound artillery superiority to force any local dukes or lords to submit to their will, setting the stage for the absolutist kingdoms to come.
Modern rocket artillery can trace its heritage back to the Mysorean rockets of Mysore. Their first recorded use was in 1780 during the battles of the Second, Third and Fourth Mysore Wars. The wars fought between the British East India Company and the Kingdom of Mysore in India made use of the rockets as a weapon. In the Battle of Pollilur, the Siege of Seringapatam (1792) and in Battle of Seringapatam in 1799, these rockets were used with considerable effect against the British. After the wars, several Mysore rockets were sent to England, but experiments with heavier payloads were unsuccessful.  In 1804 William Congreve, considering the Mysorian rockets to have too short a range (less than 1,000 yards) developed rockets in numerous sizes with ranges up to 3,000 yards and eventually utilizing iron casing as the Congreve rocket which were used effectively during the Napoleonic Wars and the War of 1812.

Napoleonic: With the Napoleonic Wars, artillery experienced changes in both physical design and operation. Rather than being overseen by "mechanics", artillery was viewed as its own service branch with the capability of dominating the battlefield. The success of the French artillery companies was at least in part due to the presence of specially trained artillery officers leading and coordinating during the chaos of battle. Napoleon, himself a former artillery officer, perfected the tactic of massed artillery batteries unleashed upon a critical point in his enemies' line as a prelude to a decisive infantry and cavalry assault.
Physically, cannons continued to become smaller and lighter. During the Seven Years War, King Frederick II of Prussia used these advances to deploy horse artillery that could move throughout the battlefield. Frederick also introduced the reversible iron ramrod, which was much more resistant to breakage than older wooden designs. The reversibility aspect also helped increase the rate of fire, since a soldier would no longer have to worry about what end of the ramrod they were using.
Jean-Baptiste de Gribeauval, a French artillery engineer, introduced the standardization of cannon design in the mid-18th century. He developed a 6-inch (150 mm) field howitzer whose gun barrel, carriage assembly and ammunition specifications were made uniform for all French cannons. The standardized interchangeable parts of these cannons down to the nuts, bolts and screws made their mass production and repair much easier. While the Gribeauval system made for more efficient production and assembly, the carriages used were heavy and the gunners were forced to march on foot (instead of riding on the limber and gun as in the British system). Each cannon was named for the weight of its projectiles, giving us variants such as 4, 8, and 12, indicating the weight in pounds. The projectiles themselves included solid balls or canister containing lead bullets or other material. These canister shots acted as massive shotguns, peppering the target with hundreds of projectiles at close range. The solid balls, known as round shot, was most effective when fired at shoulder-height across a flat, open area. The ball would tear through the ranks of the enemy or bounce along the ground breaking legs and ankles.

Modern: The development of modern artillery occurred in the mid to late 19th century as a result of the convergence of various improvements in the underlying technology. Advances in metallurgy allowed for the construction of breech-loading rifled guns that could fire at a much greater muzzle velocity.
After the British artillery was shown up in the Crimean War as having barely changed since the Napoleonic Wars, the industrialist William Armstrong was awarded a contract by the government to design a new piece of artillery. Production started in 1855 at the Elswick Ordnance Company and the Royal Arsenal at Woolwich, and the outcome was the revolutionary Armstrong Gun, which marked the birth of modern artillery. Three of its features particularly stand out.

First, the piece was rifled, which allowed for a much more accurate and powerful action. Although rifling had been tried on small arms since the 15th century, the necessary machinery to accurately rifle artillery was not available until the mid-19th century. Martin von Wahrendorff, and Joseph Whitworth independently produced rifled cannon in the 1840s, but it was Armstrong's gun that was first to see widespread use during the Crimean War. The cast iron shell of the Armstrong gun was similar in shape to a Minié ball and had a thin lead coating which made it fractionally larger than the gun's bore and which engaged with the gun's rifling grooves to impart spin to the shell. This spin, together with the elimination of windage as a result of the tight fit, enabled the gun to achieve greater range and accuracy than existing smooth-bore muzzle-loaders with a smaller powder charge.

His gun was also a breech-loader. Although attempts at breech-loading mechanisms had been made since medieval times, the essential engineering problem was that the mechanism could not withstand the explosive charge. It was only with the advances in metallurgy and precision engineering capabilities during the Industrial Revolution that Armstrong was able to construct a viable solution. The gun combined all the properties that make up an effective artillery piece. The gun was mounted on a carriage in such a way as to return the gun to firing position after the recoil.
What made the gun really revolutionary lay in the technique of the construction of the gun barrel that allowed it to withstand much more powerful explosive forces. The "built-up" method involved assembling the barrel with wrought-iron (later mild steel was used) tubes of successively smaller diameter. The tube would then be heated to allow it to expand and fit over the previous tube. When it cooled the gun would contract although not back to its original size, which allowed an even pressure along the walls of the gun which was directed inward against the outward forces that the gun's firing exerted on the barrel.