Living languages are dynamic: new words appear, the meanings of existing words change, grammar rules change etc. As a result, new languages/dialects/slangs can emerge. The relations between different languages are also dynamic: languages that share the speakers (multilingual communities) or some geographical region, can during history vary in prestige, gain or lose speakers, become lingua franca or get extinct.
We concentrate on the description of language dynamics from the complex systems theory perspective. Firstly, we discuss the motivations behind language dynamics modeling, the different sides of language complexity and the different levels of detail at which one can address the problems. Then, we present a short overview of different types of mathematical tools and models that have been introduced and used in order to address through quantitative modeling the different aspects relevant in language dynamics. The provided list is by no means complete, but should be able to give an idea about the possibilities as well as shortcomings in the mathematical modeling of the highly complex problems related to languages. We also describe how one can obtain data from social media platforms and what information and possibilities such big datasets can provide for modeling language dynamics.