The behavior of the $\phi$ meson in nuclear matter has attracted renewed interest because of (recent and future) experiments that aim to study the $\phi$ meson properties in nuclei [1-3]. Theoretically, many works have however been conducted for the $\phi$ meson at rest with respect to the nuclear medium [4-5]. In this presentation, I will review recent theoretical progress about the behavior of the $\phi$ meson in nuclear matter  and, in particular, discuss the effect of finite momentum to these results. Non-zero momentum effects will be especially relevant for future experiments, such as E16 at J-PARC, where the $\phi$ meson will normally not be measured at rest with respect to the surrounding nucleus.
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