\abstract A CP-semigroup is a semigroup $\phi=\{\phi_t: t\geq 0\}$ of normal completely positive linear maps on $\Cal B(H)$, $H$ being a separable Hilbert space, which satisfies $\phi_t(\bold 1)=\bold 1$ for all t and is continuous in the natural sense. Let dom(L) be the natural domain of the generator $L$ of $\phi$, $\phi_t=\exp{tL}$. Since the maps $\phi_t$ need not be multiplicative dom(L) is typically an operator space, but not an algebra. However, we show that the set of operators $$\Cal A=\{A\in dom(L): A^*A\in dom(L), AA^*\in dom(L) \}$$ is a $*$-subalgebra of $\Cal B(H)$, indeed $\Cal A$ is the largest self-adjoint algebra contained in $\Cal D$. Because $\Cal A$ is a $*$-algebra one may consider its $*$-bimodule of noncommutative $2$-forms $\Omega^2(\Cal A)=\Omega^1(\Cal A)\otimes_\Cal A\Omega^1(\Cal A)$, and any linear mapping $L:\Cal A \to\Cal B(H)$ has a {\it symbol} $\sigma_L: \Omega^2(\Cal A)\to\Cal B(H)$, defined as a linear map by $$\sigma_L(a\,dx\,dy)=aL(xy)-axL(y)-aL(x)y+axL(\bold 1)y, \qquad a,x,y\in\Cal A.$$ The symbol is a homomorphism of $\Cal A$-bimodules for any $*$-algebra $\Cal A\subseteq\Cal B(H)$ and any linear map $L:\Cal A\to\Cal B(H)$. When $L$ is the generator of a $CP$-semigroup with domain algebra $\Cal A$ above, we show that the symbol is negative in that $\sigma_L(\omega^*\omega)\leq 0$ for every $\omega\in\Omega^1(\Cal A)$ ($-\sigma_L$ is in fact completely positive). Examples are given for which the domain algebra $\Cal A$ is, and is not, strongly dense in $\Cal B(H)$. We also relate the generator of a $CP$-semigroup to its commutative paradigm, the Laplacian of a Riemannian manifold. \endabstract