Reductive algebraic groups are ubiquitous in areas such as geometric representation theory, geometric invariant theory, number theory... It is absolutely
outstanding that such objects have a perfectly understood classification.
I will give the main ideas of how this classification works by building intuition from the notion of a root system from the theory of semi-simple Lie algebras,
and then introduce the root datum attached to a reductive algebraic group. I will illustrate all the notions and subtleties appearing in the theory with the
groups GL_1, GL_2, SL_2 and PGL_2. If time allows I will discuss the representation theoretic aspect and introduce the so-called Langlands dual of a reductive
group and hopefully explain some properties of this (mysterious) duality.