The Cuntz semigroup of a $C*$-algebra is an important invariant in the structure and classification theory of $C*$-algebras. It captures more information than $K$-theory but is often more delicate to handle. The authors systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a $C*$-algebra $A$, its (concrete) Cuntz semigroup $mathrmCu(A)$ is an object in the category $mathrmCu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, the authors call the latter $mathrmCu$-semigroups. The authors establish the existence of tensor products in the category $mathrmCu$ and study the basic properties of this construction. They show that $mathrmCu$ is a symmetric, monoidal category and relate $mathrmCu(Aotimes B)$ with $mathrmCu(A)otimesmathrmCumathrmCu(B)$ for certain classes of $C*$-algebras. As a main tool for their approach the authors introduce the category $mathrmW$ of pre-completed Cuntz semigroups. They show that $mathrmCu$ is a full, reflective subcategory of $mathrmW$. One can then easily deduce properties of $mathrmCu$ from respective properties of $mathrmW$, for example the existence of tensor products and inductive limits. The advantage is that constructions in $mathrmW$ are much easier since the objects are purely algebraic.