In the paper slant submanifolds of a Kähler product manifold are studied. The submanifold

$M$ of a Kähler manifold

$\overline{M}$ is said to be slant if for each non zero vector

$X\in {T}_{p}M$ the angle

$\theta \left(X\right)$ between

$JX$ and

${T}_{p}M$ is independent of the choice of

$p\in M$ and

$X\in {T}_{p}M$. Consider a Kähler product manifold

$\overline{M}={\overline{M}}^{m}\times {\overline{M}}^{n}$. Denote by

$\overline{P}$ and

$\overline{Q}$ the projection operators of the tangent space of

$\overline{M}$ to the tangent spaces of

${\overline{M}}^{m}$ and

${\overline{M}}^{n}$, respectively, and put

$F=\overline{P}-\overline{Q}$. It is proved that an

$F$-invariant, slant submanifold of a Kähler product manifold

$\overline{M}={\overline{M}}^{m}\times {\overline{M}}^{n}$ is a product manifold

${M}_{1}\times {M}_{2}$ and

${M}_{1}$ (resp.,

${M}_{2}$) is also a slant submanifold of

${\overline{M}}^{m}$ (resp.,

${\overline{M}}^{n}$). It is also obtained that if

${M}_{1}\times {M}_{2}$ is the Kähler slant submanifold of

$\overline{M}$ then

${M}_{1}$ (resp.,

${M}_{2}$) is a Kähler slant submanifold of

${\overline{M}}^{m}$ (resp.,

${\overline{M}}^{n}$). In the last section several inequalities on scalar curvature and Ricci tensor for slant, invariant and anti-invariant submanifolds of a Kähler product manifold

${\overline{M}}^{m}\left({c}_{1}\right)\times {\overline{M}}^{n}\left({c}_{2}\right)$ are obtained.