Lattices are classical objects in the geometry of numbers. A lattice is a discrete subgroup of the n-dimensional vector space over the real numbers. In this thesis, we study the complexity of four classical problems from the geometry of numbers, the shortest vector problem (SVP), the successive minima problem (SMP), the shortest independent vectors problem (SIVP), and the closest vector problem (CVP). These problems can be defined for any norm on the vector space.The focus of this thesis is the algorithmic complexity of the four lattice problems described above with respect to arbitrary, especially non-Euclidean norms.Extending and generalizing results of Ajtai et al. we present probabilistic single exponential time algorithms for all four lattice problems using single exponential space. The algorithms solve SVP and restricted versions of the other problems optimally. Furthermore, the algorithms solve the general versions of SMP, SIVP, and CVP almost optimally.To obtain algorithms that solve SMP, SIVP, and CVP exactly with respect to arbitrary norms, we consider CVP in detail since there exist polynomial time reductions from SMP and SIVP to CVP which work for any norm, see [Micc08]. We will describe in this thesis a deterministic polynomially space bounded algorithm for CVP. The algorithm is based on Lenstras technique for integer programming ([Len83]).