This colloquium will explore some fundamental issues in the mathematics of plasmas, focusing on the stability and instability of solutions to Vlasov-type equations, which are crucial for describing the behavior of charged particles in a plasma. A general introduction to kinetic theory is given, making the subject accessible to a wide audience of mathematicians. Key mathematical concepts such as well-posedness, stability, and the behavior of solutions in singular limits are discussed. In addition, a new class of Wasserstein-type distances is introduced, offering new perspectives on the stability of kinetic equations.