England
Henry IV (1399-1413)
Henry had only the thinnest claim to legitimacy, being distantly related to Edward, but he was popular with the barons. He was also the richest man in England, had been wronged by Richard, and had defeated Richard in battle. That was good enough for most.
Henry had nothing but troubles as king. His chief ally was Percy, Earl of Northumberland, yet Percy himself rebelled against the new king when he did not immediately receive the favors he demanded. Percy tried to restore Richard to the throne, which was probably why the dethroned monarch was finally murdered.
This was followed in 1400 by a Welsh revolt, led by Owen Glendower. Though defeated, Glendower raised the standard of rebellion at every opportunity. In addition, the Scots caused problems repeatedly, though those rebellions were not as serious as the Welsh.
Henry himself was ill much of his life. He likely would have been in even more serious trouble except that some of the families—most notably the Nevilles (Westmorland) and the Beauforts (Somerset)—remained loyal and gave him good service.
Perhaps the wonder is that Henry survived at all, but he was forced to grant many concessions to Parliament, especially in the wake of a baronial revolt in 1403. He had to turn to Parliament repeatedly for financial support, and Parliament demanded a voice in government in the traditional form of an advisory council (1406). In 1410 and 1411, the young prince Hal was the head of this council, giving him important experience in government.
Henry Bolingbroke died in 1413. He had established a dynasty, but it was one whose legitimacy was always open to question, and he did not rule so forcefully and successfully so as to put those doubts in the past. His son would go far toward accomplishing that goal, but the barons were growing accustomed to viewing themselves as kingmakers.