Lattices are discrete periodic structures in Euclidean spaces, which are essential in many areas of mathematics and their applications. In particular, lattices are vital in number theory, discrete geometry, discrete optimization, digital communication, and other areas. Some of the most interesting lattice constructions come from various algebraic contexts, resulting in lattices with remarkable geometric properties: it appears that algebraic structure often informs the geometry. One of such properties is extremality, which ensures certain strong optimization potential, however extremal lattices are not so easy to construct. In this talk we will give a basic overview of lattice theory with a view towards optimization properties, and will exhibit some new systematic constructions of extremal lattices from surprisingly simple algebraic objects.