The purpose of this paper is twofold: Study of the mathematical background on lattice theory and its applications to cryptography. In the latter area, lattice theory has played a crucial role in the definition of new cryptosystems, in the study of cryptographic primitives and in cryptanalysis. Lattice-based public-key cryptosystems relies on computational hard problems based on the shortest vector problem (SVP) and the closest vector problem (CVP) in lattices for their security. In this paper, we present a short introduction to lattice theory and its hard problems and also we study the most important lattice-based public-key cryptosystems and digital signature together with their security analysis and some applied examples.