A semi-algebra of continuous functions is a cone $A$ of continuous real functions on a compact Hausdorff space $X$ such that $A$ contains the products of its elements. A cone $A$ is said to be of type $n$ if $f\in A$ implies $f^n(1 + f)^{-1} \in A$. Uniformly closed semi-algebras of types 0 and 1 have long been characterized in a manner analogous to the Stone--Weierstrass theorem, but, except for the case when $A$ is generated by a single function, little has been known about type 2. Here, progress is reported on two problems. The first is the characterization of those continuous linear functionals on $C(X)$ that determine semi-algebras of type 2. The second is the determination of the type of the tensor product of two type 1 semi-algebras. 1991 Mathematics Subject Classification: 46J10.