The idea of text messaging goes back almost to the first days of mobile phone development. However, the idea of the SMS (short message service) was mainly developed in 1984 by Friedhelm Hillebrand and Bernard Ghillebaert, working for the DF Project Group. This was a Franco-German group that wanted to find a way to standardise GSM - Global System Mobile Communication.

Because the new system was short, with a limited number of characters per messaged, it was possible to use it in existing mobile systems. At the beginning, the developers wondered if such a messaging service would be popular with mobile users, as nothing like this had been tried before. The mainly used messaging service at that time was Telex.

However, SMS was standardised in 1985, the first message was sent in 1992 and it quickly became of major importance in the field of communications. You can read a detailed history here.