This paper presents a polynomial-time approximation scheme (PTAS) for the Euclidean Traveling Salesman Problem (TSP) in fixed dimensions. For any fixed constant c > 1, the algorithm finds a (1 + 1/c)-approximation to the optimal TSP tour in O(n(log n)^{O(c)}) time for points in R², and O(n(log n)^{(O(√d c))^{d-1}}) time for points in R^d. The algorithm can be derandomized, but this increases the running time by a factor of O(n^d). The previous best approximation algorithm for TSP, due to Christofides, achieves a 3/2-approximation in polynomial time. The paper also provides similar approximation schemes for other NP-hard Euclidean problems, including Minimum Steiner Tree, k-TSP, and k-MST. These algorithms achieve a (1 + 1/c)-approximation with a running time that includes an additional multiplicative factor of k. The paper also presents efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time. All the algorithms work with almost no modification when distance is measured using any geometric norm, such as ℓ_p for p ≥ 1 or other Minkowski norms. They also have simple parallel (i.e., NC) implementations. The algorithm for TSP is based on recursively partitioning the plane using a randomized variant of the quadtree. The key idea is that a (1 + 1/c)-approximate TSP tour crosses each line of the partition at most O(c) times. This allows the use of dynamic programming to find the optimal tour. The algorithm is proven to have a (1 + 1/c)-approximation with high probability. The paper also discusses the implications of the PTAS for other geometric problems and the practicality of the algorithm. It notes that while the algorithm is theoretically efficient, its practical implementation may be slow for moderate values of c. However, the algorithm provides a way to decompose TSP instances into smaller, independent subproblems, which may be useful for parallelization. The paper also highlights the potential theoretical contributions of the algorithm to the understanding of local-exchange heuristics on Euclidean instances.This paper presents a polynomial-time approximation scheme (PTAS) for the Euclidean Traveling Salesman Problem (TSP) in fixed dimensions. For any fixed constant c > 1, the algorithm finds a (1 + 1/c)-approximation to the optimal TSP tour in O(n(log n)^{O(c)}) time for points in R², and O(n(log n)^{(O(√d c))^{d-1}}) time for points in R^d. The algorithm can be derandomized, but this increases the running time by a factor of O(n^d). The previous best approximation algorithm for TSP, due to Christofides, achieves a 3/2-approximation in polynomial time. The paper also provides similar approximation schemes for other NP-hard Euclidean problems, including Minimum Steiner Tree, k-TSP, and k-MST. These algorithms achieve a (1 + 1/c)-approximation with a running time that includes an additional multiplicative factor of k. The paper also presents efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time. All the algorithms work with almost no modification when distance is measured using any geometric norm, such as ℓ_p for p ≥ 1 or other Minkowski norms. They also have simple parallel (i.e., NC) implementations. The algorithm for TSP is based on recursively partitioning the plane using a randomized variant of the quadtree. The key idea is that a (1 + 1/c)-approximate TSP tour crosses each line of the partition at most O(c) times. This allows the use of dynamic programming to find the optimal tour. The algorithm is proven to have a (1 + 1/c)-approximation with high probability. The paper also discusses the implications of the PTAS for other geometric problems and the practicality of the algorithm. It notes that while the algorithm is theoretically efficient, its practical implementation may be slow for moderate values of c. However, the algorithm provides a way to decompose TSP instances into smaller, independent subproblems, which may be useful for parallelization. The paper also highlights the potential theoretical contributions of the algorithm to the understanding of local-exchange heuristics on Euclidean instances.