In recent years, significant progress in researching and building quantum computers has been made. The existence of such threatens the security of many modern cryptographic systems. This affects, in particular, asymmetric cryptography, i.e. KEMs (key encapsulation mechanisms) and digital signatures. A fully-fledged quantum computer would be able to efficiently solve a distinct set of mathematical problems like integer factorization and the discrete logarithm, which are the basis for a wide range of cryptographic schemes. Therefore, the need for novel mathematical problems which are resistant to the computing power of quantum computers arises. In 2016, NIST announced an open competition with the goal of finding and standardizing suitable algorithms for quantum-resistant cryptography (as it also did for AES and SHA-3). The standardization effort by NIST is aimed at post-quantum secure KEMs and digital signatures. This process is currently in its fourth round of candidate selection (June 2023). In this article, two of the to-be-standardized algorithms, Kyber and Dilithium, are presented and some of their mathematical details are outlined. Both algorithms are based on so-called lattices and the thereupon constructed »Learning with Errors«, which we will get to know in the following.