This thesis examines two topics in approximation algorithms. Mechanism design considers algorithmic problems in which agents behave based on selfish needs, rather than the will of the mechanism. For the knapsack problem, a number of approximate mechanisms are described that guarantees truthful agent behavior, including an FPTAS recently constructed by Alberto Marchetti-Spaccamela. Results relating truthfulness to the priority algorithm framework of Borodin, Nielsen and Rackoff are shown.A formal algorithmic model, called the stack algorithm, is defined, that captures the behavior of the local ratio method. The bandwidth problem is defined, and limitations are shown on the approximation power of the stack algorithm in a number of variations, including 2 machine scheduling. For covering problems, approximation lower bounds are shown for the Steiner tree and set cover problems.