To my untrained eye, the bow seems more or less the same basic shape and construction since... well forever. I assume there must have been some big and important changes in material, especially in strings or fletching, but was there any big evolution during the Middle Ages? Did the idea of composite bows spread very far?
A wooden self-bows is a simple thing in many ways: a tapered close-to-straight piece of wood, and a string. The basic shape was already established in the earliest bows we have found (Mesolithic examples from northern Europe). Excellent material for strings such as linen and sinew (and silk, in some parts of the world) were available in ancient times (linen was used for fibre long before the domestication of flax, possibly as long ago as 30,000 years or more). The materials for arrow shaft and fletching are also long-known: wood, bamboo, cane, reeds, and feathers.
Going beyond the self bow, we have the composite bow. The details of the origin of the composite bow are not certain, but the composite bow was widespread across Eurasia by the mid-2nd millennium BC, and widely used alongside the chariot. Thus, it may have been in in use alongside the chariot in the early stages of the Indo-European expansion beginning c. 2000BC. Composite bows were used across Europe in Roman times, and continued to be used in Spain and Eastern Europe through much of the Medieval period, and were known and occasionally used other parts of Europe (e.g., in Germany). Composite crossbow prods were widely used in the later Medieval period (see below), so the idea of the composite construction was known, even if the composite bow wasn't used.
Both wooden self-bows and composite bows were highly developed long before the Medieval period, making excellent use of the materials available. In addition, an essential part of the system - the human archer - remained basically unchanged, and limited possible increases in power. Significant further development would require either new materials or a radical new design, both of which came with modern technology. New materials included plastics, fibreglass, carbon fibre, modern adhesives. new designs, in part enabled by new materials included the compound bow, with pulleys, cams, and multiple strings, and also accessories such as stabilisers and sights.
What development was possible during the Medieval period? It has been suggested that armour become better and/or more common resulted in an arms race pushing the draw weights of military bows to very high levels. Alas, we have a serious shortage of bows from the Medieval period, and little concrete idea about the evolution of draw weights. Chinese literature suggests that high draw weight bows were known in antiquity. From Egypt, the Armant stela, which says of the archery skills of Thutmose III:
When he shoots at a copper target, all wood is splintered like a papyrus reed. His Majesty offered an example thereof in the temple of Amun, with a target of hammered copper of three digits in thickness; when he had shot his arrow there, he caused protrusion of three palms behind it, so as to cause the followers to pray for the proficiency of his arms in valour and strength. I am telling you what he did, without deception and without lie, in front of his entire army, and there is no word of exaggeration therein.
also describes very powerful bows (which must, however, be treated cautiously, as this stela glorifies a king and his military exploits, and while the text claims there is no exaggeration, this shouldn't be taken literally). There was possibly an increase in draw weights from about 50lb to over 100lb.
One change in the availability of materials was increased availability of iron and steel, and more consistent better quality steel. Iron and steel became more widely used for arrowheads, replacing heads of bone and horn in some cases. Iron/steel arrowheads were already in use in antiquity, so this wasn't new technology, but just an increasingly cheap technology.
One new type of bow did develop, or at least became more common: the Indian steel bow:
These steel bows do not seem to have been particularly powerful (nor particularly weak). They have relatively heavy limbs, and are less efficient (i.e., more the energy is lost in moving the limbs and less transferred to the arrow). The have the advantage of being little affected by the weather, and might have been mostly made for long-term storage in arsenals, and for the defence of fortresses, where they could usefully shoot very heavy arrows from the walls.
Unlike the bow, the crossbow evolved considerably in Europe over the Medieval period, from simple hand-spanned bows with wooden prods through powerful crossbows with composite prods and various spanning devices, to powerful crossbows with steel prods. If you are interested in crossbow evolution, there is an excellent thesis by u/Valkine which will provide you with some fun reading:
Medieval Chinese crossbow show less evolution, since (a) powerful crossbows with sophisticated triggers already existed in antiquity, and (b) Chinese crossbows used long power strokes (and therefore long prods) for power, and remained possible to span without mechanical aid throughout this period.
References:
Amant stela text: https://mjn.host.cs.st-andrews.ac.uk/egyptian/texts/corpus/pdf/ArmantTuthmosisIII.pdf
On Indian steel bows:
D. Elmy, "D. Elmy": http://margo.student.utwente.nl/sagi/artikel/steelbow/steelbow.html
T. A. Nieminen, "The Asian war bow": https://arxiv.org/abs/1101.1677
