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Muktze (also spelled muktzah sometimes) is a really comlicated subject connected to the laws of shabat. Basically, there are items that have no use on shabat, e.g., a pen, because you are not allowed to write on shabat. So the rabbis introduced a "fence" around the Torah that you are not allowed to move such an item on shabat. The idea is to prevent you from accidentally taking out the pen to write something down because you carry it around in you pocket. This is the rough idea, the details are very complicated and I don’t really know them.

What I want to do in this post is address a common misconception, namely that muktze means something may not even be touched. I have had this discussion several times. At least three times with my chavruta, even though we have learned the little that I know together. And he is not alone, I have found several internet places where (lay) people tell others something along the lines of "muktze, meaning they cannot be touched" or even official how-to-guides to shabat that are ambiguous and over-simplified and "explain" muktze with one sentence similar to "In addition to those mentioned above [the categories of forbidden work on shabat], two other important categories which are not permitted are using or touching items that are considered muktzah and carrying outdoors." (Chabad: The Shabbat Laws).

So, to set this straight: Muktze refers to items that have no use on shabat and may not be moved, but they may be touched on shabat. The only time they may not be touched is if the touching may cause them to move (e.g., a very light object or a round object).

Here are some sources that you can use to confirm this (pretty random, what I was able to find fast):

All ”muktzeh” (articles) are only forbidden to be handled, but touching them, if it doesn’t cause them to move, is allowed.
(Kitzur shulchan aruch 88:12)

The Rama tells us that muktze may be touched but not moved. This, however seems to contradict another halacha, [6] which says that one may cover a muktze as long as one does not touch it while doing so. The Mishna Berura [7] reconciles the two by saying that the latter halacha is referring to covering an egg or something round. Since an egg is oval shaped, touching it will definitely move it, and therefore it may not be touched. Other muktze items that will not move when touched may be touched.
(Paraphrase on Mishna Berura 310:6 [footnote 6] and 310:22 [footnote 7] by Rabbi Dovid Ostroff)

More convincing may be the lack of even the word "to touch" in the places that otherwise discuss muktze, such as the Rambam’s Mishne Torah, Hilchot Shabat chapters 25 and 26, or any more in-depth introduction to the topic, e.g., Rabbi David Shure’s introduction to The thirty-nine works of the sabbath. Feel free to comment with other good sources!