Many NP-hard optimization problems that are equivalent under Karp reductions behave very differently when we ask for polynomial-time approximations. This paper studies the complexity class APX, a subclass of NPO consisting of optimization problems that admit polynomial-time algorithms with constant-factor approximation guarantees. We first review the formal structure of NPO, define L-reductions and PTAS-reductions, and prove that L-reductions preserve polynomial-time approximation schemes (PTAS). Using the Probabilistically Checkable Proofs (PCP) theorem as a starting point, we present self-contained L-reductions establishing the APX-completeness of Max-3SAT, Max-2SAT, Max-NAE-3SAT, and Max-Cut, and we situate APX within the broader hierarchy FPTAS, PTAS, APX, logAPX, and polyAPX. We then examine structural consequences of the polynomial hierarchy for self-improvability and APX-intermediate problems such as Min-Bin-Packing, and review how APX-hardness arises in Bayesian optimal mechanism design for three-bidder single-item auctions. Turning to algorithm design, we analyze three core techniques across the APX and logAPX boundary: linear programming relaxation with Raghavan-Thompson randomized rounding for multicommodity congestion minimization, Goemans-Williamson semidefinite programming rounding for Max-Cut, and greedy, dual-fitting, and randomized rounding methods for Min-Set-Cover. We close by discussing bounded-query approximation oracles and recent constant-factor and sub-3/2 advances on the asymmetric and metric traveling salesman problems.

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