A strict quantization of a compact symplectic manifold $S$ on a subset
$I\subseteq\R$, containing 0 as an accumulation point, is defined as a
continuous field of $C^*$-algebras $\{A_{\hbar}\}_{\hbar\in I}$, with
$A_0=C_0(S)$, and a set of continuous cross-sections $\{Q(f)\}_{f\in
C^{\infty}(S)}$ for which $Q_0(f)=f$. Here $Q_{\hbar}(f^*)=Q_{\hbar}(f)^*$ for
all $\hbar\in I$, whereas for $\hbar\to 0$ one requires that
$i[Q_{\hbar}(f),Q_{\hbar}(g)]/\hbar\to Q_{\hbar}(\{f,g\})$ in norm. We discuss
general conditions which guarantee that a (deformation) quantization in a more
physical sense leads to one in the above sense.
Using ideas of Berezin, Lieb, Simon, and others, we construct a strict
quantization of an arbitrary integral coadjoint orbit $O_{\lm}$ of a compact
connected Lie group $G$, associated to a highest weight $\lm$. Here $I=0\cup
1/\N$, so that $\hbar=1/k$, $k\in\N$, and $A_{1/k}$ is defined as the
$C^*$-algebra of all matrices on the finite-dimensional Hilbert space
$V_{k\lm}$ carrying the irreducible representation $U_{k\lm}(G)$ with highest
weight $k\lm$. The quantization maps $Q_{1/k}(f)$ are constructed from coherent
states in $V_{k\lm}$, and have the special feature of being positive maps.