The ulnar artery, [a. ulnaris], extends from the point of bifurcation of the brachial just indicated, along the inner side of the fore-arm, into the palm of the hand, where, joining a branch of the radial, opposite the muscles of the thumb, it forms the superficial palmar arch. In this course it inclines first downwards and inwards, describing a slight curve, the convexity of which is directed inwards, and passes under cover of the superficial muscles arising from the inner condyle of the humerus, viz., the pronator teres, flexor carpi radialis, palmaris longus, and flexor sublimis, until it reaches the flexor carpi ulnaris, about the junction of the upper with the middle third of the fore-arm ; at this point the artery comes into contact with the ulnar nerve, (previously separated from it by a considerable interval,) and, changing its direction, descends vertically with the nerve towards the inner border of the palm of the hand. Guided as it descends by the tendon of the flexor ulnaris muscle, along the radial border of which it is now placed, the ulnar artery reaches the outer or radial side of the pisiform bone, where, still accompanied by the nerve, it passes over the cutaneous surface of the anterior annular ligament of the wrist into the palm of the hand. Its disposition in the hand will be separately described.

In the first half of its course through the fore-arm, the artery is deep-seated, being covered by the muscles arising from the inner condyle of the humerus which have been already enumerated. About the middle of the fore-arm it is only slightly overlapped by the flexor carpi ulnaris; but below that, it becomes comparatively superficial, being covered only by the skin, the fascia of the fore-arm and a thin layer of membrane by which the vessel is bound down to the muscle beneath. At first the ulnar artery rests on the insertion of the brachialis anticus into the coronoid process of the ulna ; then on the flexor profundus in the rest of the fore-arm ; and lastly, on the annular ligament of the carpus. Below the point at which it emerges from under the flexor carpi ulnaris, (or a little below the middle of the fore-arm,) the tendon of that muscle is on its inner or ulnar side.

Nerves

The median nerve lies immediately on the inner side of the ulnar artery at its origin, but the nerve soon passes over the vessel, and is then separated from it by the second head of the pronator teres muscle. As the ulnar nerve descends behind the inner condyle of the humerus, it is separated from the ulnar artery by a considerable interval at the upper part of the fore-arm ; but as the artery inclines inwards, it approaches the nerve, and is accompanied by it in the lower half of its course — the nerve lying close to its inner side. A small branch of the ulnar nerve descends upon the lower part of the vessel.

Veins

Two veins (vense comites), which have the usual arrangement of such veins, accompany the ulnar artery.

On the wrist, the ulnar artery rests on the anterior annular ligament, and is covered by the skin and fascia. The pisiform bone is to its inner side ; the ulnar nerve is also on the same side, but somewhat behind the artery. 

From Quain's anatomy.