The History of the LLL-Algorithm -- Hermite#x2019;s Constant and Lattice Algorithms -- Probabilistic Analyses of Lattice Reduction Algorithms -- Progress on LLL and Lattice Reduction -- Floating-Point LLL: Theoretical and Practical Aspects -- LLL: A Tool for Effective Diophantine Approximation -- Selected Applications of LLL in Number Theory -- The van Hoeij Algorithm for Factoring Polynomials -- The LLL Algorithm and Integer Programming -- Using LLL-Reduction for Solving RSA and Factorization Problems -- Practical Lattice-Based Cryptography: NTRUEncrypt and NTRUSign -- The Geometry of Provable Security: Some Proofs of Security in Which Lattices Make a Surprise Appearance -- Cryptographic Functions from Worst-Case Complexity Assumptions -- Inapproximability Results for Computational Problems on Lattices -- On the Complexity of Lattice Problems with Polynomial Approximation Factors.