The Biceps brachii crosses both the elbow and shoulder joints. Its action on the shoulder joint is very weak flexion.
The Brachialis acts to flex the elbow whether in pronation or supination, along with Biceps Brachii. As Brachialis is attached to the Ulna, which cannot rotate, it is the only true flexor of the elbow.
The Brachioradialis muscle acts to supinate the forearm from a pronated position, when it flexes the elbow. When starting in a supinated position, it acts to pronate the hand as it flexes the elbow.
Flexor Carpi Radialis crosses the elbow joint and so is also a weak elbow flexor.
Flexor carpi ulnaris is one of the wrist flexor muscles of the front (palmar) aspect of the forearm. It also helps extensor carpi ulnaris to ulnar deviate the wrist.
Flexor Digitorum Superficialis is sometimes also known as Flexor Digitorum Sublimis. It is one of the wrist flexor muscles found in the palm side of the forearm and wrist.