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.
Since the Suzuki and Ree groups have BN-pairs, it is popular with finite group theorists to use this viewpoint in studying them (simplicity, etc.).