Please don't give up on me if this composer stuff is boring you. As soon as it's over with, I can start dealing with the good stuff - like, for example, what certain symphonies mean to me!
Rameau (Jean-Philippe) dominated the world of French opera for some time, but by the end of the 18th century his operas had fallen into obscurity. His music was revived a century later, and his innovations in the dramatic form of French opera influenced many later composers. He also wrote a Treatise on Harmony that can be seen as the foundation of the modern science of music theory. Today one can still hear performances from his three books of harpsichord pieces, and some of his operas have been revived recently, including Hippolyte et Aricie.
Rautavaara (Einojuhani) is the leading living Finnish composer, writing a great deal of music that is accessible to the general public. His works include symphonies (8), operas, chamber music, concertos, piano music, choral works, and orchestral works, including some that combine live musicians with tape recordings. His seventh symphony, titled Angel of Light, is part of a series of works based on visions and revelations he experienced as a child.
Reubke (Julius) was one of the brief, blazing lights of the fine-art-music firmament. The son of an organ builder, Reubke became Liszt's favorite pupil and managed, before his consumptive death at age 24, to write a world-class Piano Sonata and an Organ Sonata regarded today as one of the greatest masterpieces ever written for that instrument. These two works, besides a few minor pieces, are enough to earn him a place in music history, even though their style is decidedly influenced by Liszt. Imagine what would have happened had he lived to develop his own unique style!
Rodrigo (Joaquín) was a blind Spanish pianist. His compositions include Concierto de Aranjuez, one of the most important guitar concertos. His other guitar concertos include Fantasia para un Gentilhombre, written for Andrés Segovia, and the Concierto de Andaluz for four guitars and orchestra. He also wrote concertos for piano, cello, violin, flute, and harp. He died in 1999, covered with awards and honors, at the age of 97.
Rore (Cipriano de) was a Flemish composer of the 16th century. Though he wrote sacred motets and masses, he is best known for his Italian madrigals (secular songs), written in a very sophisticated style.
Rorem (Ned) is an American composer and author of the celebrated Paris Diary, a kiss-and-tell account of his relationships with other men in the musical world. He also wrote three symphonies, concertos for piano, flute, and violin, a double concerto for violin and cello, an opera version of Our Town, and many well-respected songs.
Roussel (Albert) was a French composer around the turn of the 20th century. His musical style shifted from lush, gushy romanticism to lean, mean, modern neo-classicism; I personally prefer the latter style. His third symphony kicks derriere. He wrote four symphonies in all (No. 4 is also recommended), plus concertos, ballets (including The Spider's Feast), orchestral suites, chamber music, etc., etc., etc.
Rutter (John) is a British composer known chiefly for his choral music, including arrangements of many Christmas carols, plus his own Requiem and Mass of the Children, the latter written in memory of his own son; he even wrote a children's opera about the Gunpowder Plot, titled Bang! Rutter's choral music is currently sung by church and school choirs throughout the English-speaking world.