To supplement Torsten's account, the original Suzuki groups of type $C_2$ in characteristic 2 resulted from a purely group-theoretic investigation but were then recovered in the algebraic group setting.   The Ree groups of types $F_4, G_2$ in respective characteristics 2, 3 were constructed inside the Chevalley groups of these types but also recovered in a uniform way by Steinberg in *Endomorphisms of algebraic groups* (AMS Memoir).  There is also a full account in my recent LMS Lecture Note volume *Modular Representations of Finite Groups of Lie Type* (Cambridge, 2006).   The algebraic group viewpoint is outlined by Torsten.  The Suzuki and Ree groups don't arise from the split vs. quasisplit classification over finite fields, but rather involve Chevalley's *special isogenies* which interchange root lengths while using a finite field automorphism.    The orders of the finite fields one starts with are the *odd* powers of 2, 3 respectively.   But notation is tricky, since some people like to express things in terms of square roots to make the finite group orders resemble those of the corresponding split groups.