Henry Sire recently published a book called The Knights of Malta: A Modern Resurrection about just this topic (for the Knights Hospitaller, at least).

A brief background: The Knights Hospitaller staged a centuries-long retreat from the Holy Land, ending up in Cyprus (as the Knights of Cyprus) and Rhodes (ditto), before they obtained the island of Malta in 1530 from the King of Spain (rented for an annual tribute of one Maltese Falcon.)

But in 1798 Napoleon redrew the map of Europe and the Knights lost Malta--they also lost all their priories in France, almost all in Europe, and wound up in Russia for a bit because the Tsar wanted to be the head of the order... finally they wind up in Rome. They worked in hospitals and care for the sick, as they had earlier. (There are also Protestant orders, that's a bit beyond our scope.) So they were now officially the Sovereign Military Order of Malta.

Men were still joining the order in the 19th and 20th century, but the Knights of Malta still had major nobility requirements to become a fully professed Knight--I believe these weren't removed until the 1990s. According to Sire, it was not unknown for older Italian and German noblemen, who had been widowed, to join the Order as a sort of retirement community. Apparently they spoke English amongst themselves, read P. G. Wodehouse, and referred to one particularly bustling Prior as "Mother" behind his back.

This contrasts with the other work of the Order in founding hospital services and ambulance services in other countries like Ireland. I believe these services took charge of transporting sick pilgrims to Lourdes, among other work. But one didn't need to be noble, or commit to a religious life, to join these auxiliary orders. Many people still work for the Knights of Malta in some capacity, but without taking religious vows or joining as any sort of knight.

In any case the Order has been given Observer status by the UN, and can issue passports, etc.

I believe that's one of the main tensions now--how the Order will change and grow to function in a more democratic, less religious world.

How the military orders kept up with the times varied pretty greatly from order to order, but all of those still around today have abandoned any military role. The Spanish military orders, the Knights of Santiago, Order of Calatrava, Order of Alcantara, and others, secularised. Throughout the late medieval period they increasingly served as royal commanders and also participated heavily in Portugal's first expeditions along the West African coast. In 1523 all the Spanish orders were taken under royal control, they abandoned their monastic vows and lifestyle, and eventually became the monarchical Spanish orders they are today, and the same process happened to the Portuguese military orders.

The Hospitallers are still around today, as the Sovereign Military Order of Malta. Their later history was a bit more complicated. In England, Ireland, Wales, and Scotland they had no choice about how to modernise, they were suppressed at the Reformation. In Germany the Bailliwick (an administrative division of the Hospitallers) of Brandenburg broke away and converted to Lutheranism. But the bulk of the Order, now based on Malta, remained Catholic and conducted pirate hunting campaigns (and some piracy of their own) against Muslim ships in the Mediterranean. They retained their lands in France, the Iberian Peninsula, Italy, and much of Germany. They even bought a few Caribbean colonies, including St Barthelemy, from France in the 1650s, but sold them soon after. The Hospitallers' naval success became quite famous. From the mid 1600s until 1748 the Order trained and commanded many of the French navy's galleys. From the 1760s until 1796, the Hospitallers also hosted and trained Russian naval officers. The Order continued these naval campaigns all the way up to 1798 when Napoleon conquered Malta. After this some of the Order relocated to Russia, which had offered them sanctuary. Eventually the Order settled in Rome in 1834, where they are still headquartered, though they also now have some presence in Malta again.

The Teutonic Order's later history was also complicated, but from it can be traced the foundations of modern Germany. From 1410 the Order's state in the Baltic had been without a purpose. The pagan Lithuanians had long since converted to Christianity and that year a combined Polish-Lithuanian force had crushed the Order at the Battle of Tannenberg. The Order continued to raid after this date, but crusaders no longer joined them from abroad. Attempts were made to relocate the Knights to Transylvania to fight the encroaching Turks, but this failed when it was made conditional on them abandoning Prussia. From 1454 there was civil war in the Order's Prussian lands and war soon broke out with Poland. This ended in 1466 with the Prussian branch of the Order accepting the Polish king as their overlord. In 1524 the master of the Order met Martin Luther at Nuremburg and converted to Protestantism. Those knights that did not agree returned to Germany. The following year he signed the Treaty of Cracow with the Polish king, turning the Order State into the secular duchy of Prussia. His heirs held the duchy until 1618, when it was inherited by the Hohenzollerns of Brandenburg who would eventually unite Germany into a single state. The Livonian branch of the Order, in what is now Latvia and Estonia, survived a little longer. They engaged in a series of wars with the Russians finally ending in 1562 when the Order handed over its territories Poland and disbanded. The last master of Livonian knights converted to Protestantism and ruled the south-west of Livonia as a new Duchy of Courland.

The Teutonic Knights continued to have a military role after the loss of Prussia and Livonia. From 1695 the Order funded and led the Hoch-und-Deutschmeister regiment of the Austro-Hungarian army until the fall of that empire in 1918. The Order survives today but is now solely charitable, though it retains some of the military ceremony.

The smaller military orders faded away from military life. The Knights of St Thomas of Acre, a small English order, abandoned its military role soon after the loss of Acre in 1291. They focused on hospital work in England and were eventually suppressed at the Reformation. The Order of St Lazarus, the Leper Knights, split into multiple branches. The English knights broke away, demilitarised, and were suppressed at the Reformation. In Germany the Order merged with the Hospitallers in 1490. The Italian Knights were formed into the Order of St Maurice and St Lazarus, which took up anti-Muslim naval campaigns like the Hospitallers. The Order in France was refounded as the Order of Mont Carmel and St Lazarus in 1609. Today the Order is scattered into many multiple versions, Scotland alone has three.

The Templars were suppressed in 1312 and no direct successor order survives today. There are modern orders that claim to be a successor of the Templars, but there is no evidence to support such a connection.

New orders were set up or proposed in the early modern period, such as the Order of St Stephen, founded in 1561. These often adapted the traditional model of a military order to remain relevant. The Stephen Knights were allowed to marry, unlike the Hospitallers or Teutonic Knights. There were also benefits to joining the Order. Brethren could purchase status as a noble for themselves and their heirs by joining for three years and paying to set up a new commandery (an estate) for the Order. This was obviously quite popular as at least 695 of these commanderies were established between 1563 and 1737.

Sources:

Anthony Luttrell, ‘The Military Orders, 1312-1798’, in Jonathan Riley-Smith (ed.), The Oxford Illustrated History of the Crusades (Oxford, 1995), pp. 326-64.

David Marcombe, Leper Knights: The Order of St Lazarus of Jerusalem in England, c.1150-1544.

Helen Nicholson, The Knights Hospitaller (Woodbridge, 2001).

Desmond Seward, The Monks of War (London, 1972).