Since the time of the Rishonim it has been customary to add psalms to Pesukei De-zimra on Shabbat morning. The added psalms have either creation or the giving of the Torah as their theme, for Shabbat commemorates the creation of the world and the Torah was given on Shabbat. Before Yishtabaĥ, we add Nishmat Kol Ĥai – a prayer of wondrous praise that mentions the Exodus and thus links to Shabbat, which commemorates the Exodus (Tur §281; Levush; see Peninei Halakha: Shabbat 4:4).
After Shaĥarit we read from the Torah and recite a haftara from the works of the Prophets. We already learned about Torah reading (20:15), and, as noted, on Shabbat there are seven aliyot. Here we will add that the purpose of the reading on Shabbat is to complete the entire Torah. During the time of the Talmud, it was customary in Eretz Yisrael to complete the Torah every three years while in Babylonia it was customary to complete the Torah annually, as is the custom today. If on one Shabbat the Torah portion was not read, two portions are read on the following Shabbat so as to make up for the missed portion (SA 135:2).
Following the reading of the Torah and the haftara, Musaf is recited. It corresponds to the additional offerings that we were commanded to bring to sanctify Shabbat. We similarly recite Musaf on every holiday that we are commanded to sanctify through additional offerings (see below, section 6).