Improvisation is quite a lost art in the classical world for quite a long time. I do not know of any classical music program that includes improvisation as a required subject in its syllabus. I would be glad to know if such a program exist somewhere.
Classical composers like Bach, Mozart, Clementi, Schubert, Chopin, Liszt, Brahms, Debussy were some of the most renown improvisers of their time. It is said that the genius of Albert Einstein was triggered by his improvising on the violin since he was young.
Practitioners of improvisation today are those from the contemporary music world, especially jazz musicians. Back in the 60s and 70s, popular music did feature a solo (improvisation) that is usually played by the guitarist or keyboardist.
Personally, improvisation is a natural progress in musical development when one seeks to create music instead of just playing prepared music. To improvise requires many facilities to come together and these facilities have to be developed from the first day of music lessons. Developing a good appreciation in pitch and an understanding of harmony is primary.
Perhaps not all, but most music can be improvised on. One only has to know enough to do the right thing. In the attached youtube clip, the students are merely trying their hand at improvising for the first time. They are kids who began learning music about 2 to 3 years ago.
Using a very familiar song from the book they began learning with is important. In this case, the choice was Mary Had A Little Lamb. In this first workshop, they are learning to embellish the melody through hearing and imitating the idea.
In the second clip below, the student is improvising to an accompaniment of the same song. It was solely the creativity of the student and no written improvisation of the song was provided. I only had to demonstrate a few versions of improvisation of the song for the students listening appreciation. After which, I invited them to try their hand at it and this student produced the best improvisation during the workshop.