This paper presents a method using Lovász' lattice reduction to find a lattice point near a given point in R^d, within a factor of c^d (c=const.). It shows that two straightforward algorithms can achieve this when applied to a lattice given by a Lovász-reduced basis. The paper also proves a geometric property of Lovász-reduced bases: a lower bound on the angle between any basis vector and the hyperplane generated by the others. This leads to an application in nonhomogeneous simultaneous diophantine approximation, optimal within a factor of C^d. Another application is improving the Grötschel-Lovász-Schrijver version of H. W. Lenstra's integer linear programming algorithm. The algorithms run in polynomial time when applied to rational input vectors. The paper discusses the importance of nonhomogeneous diophantine approximation and the approximately nearest lattice point problem. It also mentions that the nearest lattice point problem is NP-hard, but efficient algorithms can find a point within C^d times the distance from the nearest one. The paper also raises a problem about whether an oracle that solves the nearest lattice point problem can be used to solve the shortest vector problem within a certain factor. The author thanks Vera Sós and László Lovász for their contributions. The paper highlights the significance of Lovász' lattice reduction algorithm in solving various diophantine problems and its practical performance.This paper presents a method using Lovász' lattice reduction to find a lattice point near a given point in R^d, within a factor of c^d (c=const.). It shows that two straightforward algorithms can achieve this when applied to a lattice given by a Lovász-reduced basis. The paper also proves a geometric property of Lovász-reduced bases: a lower bound on the angle between any basis vector and the hyperplane generated by the others. This leads to an application in nonhomogeneous simultaneous diophantine approximation, optimal within a factor of C^d. Another application is improving the Grötschel-Lovász-Schrijver version of H. W. Lenstra's integer linear programming algorithm. The algorithms run in polynomial time when applied to rational input vectors. The paper discusses the importance of nonhomogeneous diophantine approximation and the approximately nearest lattice point problem. It also mentions that the nearest lattice point problem is NP-hard, but efficient algorithms can find a point within C^d times the distance from the nearest one. The paper also raises a problem about whether an oracle that solves the nearest lattice point problem can be used to solve the shortest vector problem within a certain factor. The author thanks Vera Sós and László Lovász for their contributions. The paper highlights the significance of Lovász' lattice reduction algorithm in solving various diophantine problems and its practical performance.