What era do these pieces belong to? Links are to orchestral recordings by André Rieu.
They are neither classic nor baroque, but are they romance ?
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The "Second Waltz" is an arrangement of "Waltz No. 2" (from the Jazz-Suite), composed by Dmitri Shostakovich (1906-1975). It is a very non-typical piece (light in tone) of him. He was a 20th century composer but this piece may be considered a pastiche on the works of Johan Strauss Jr. (1825-1899) (not to be confused with Johann Strauss (1804-1849)!) who mostly composed light classical dance music (as opposed to more serious music meant to listen to). The genre/era of these kind of waltzes is romantic.
The André Rieu recording entitled "Second Waltz" that you have linked to is from an album called Forever Vienna. It is a waltz, composed by Russian composer Dmitri Shostakovich (1906-1975). It may be considered a pastiche on the works of the famous Austrian composer Johan Strauss II (1825-1899), so you could consider this a more recent piece meant to sound like something from the Romantic style period. However, as with the other pieces you are asking about, André Rieu has created his own contemporary orchestral arrangements in his own style.
"Amazing Grace" is a sacred Christian hymn which is sung by the congregation in church. It was written in 1779 by John Newton, who was English. It has become tremendously popular all over the world ever since the that time. There have been so many different arrangements and recordings in so many different musical styles that it would be impossible to list them.
A bolero, generally speaking, is a particular type of dance, and dance music. It first appeared in the late 1700s in Spain, and was also popular in Cuba. In 1928, French composer Maurice Ravel (1875-1937) wrote a long, slow orchestral piece called Bolero to accompany a ballet. The original version is about 15 minutes long. This would be considered to be in the style period referred to by some as "Post-Romantic" or "Modern" or simply "early 20th-century". On the link you have provided, André Rieu performs a much shorter arrangement of Ravel's original.