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Zadoff-Chu (ZC) sequences are an important manifestation of spread spectrum in modern cellular systems, including LTE and 5G NR. They have to some extent displaced PN and Walsh sequences which were the mainstays of 3G cellular (WCDMA and cdma2000) and the 2G-era IS-95. ZC sequences are complex sequences with unit amplitude and particular phase shifts, as opposed to Walsh and PN codes which are real and binary valued, most commonly $\pm1$. ZC sequences have a number of remarkable and desirable properties that we define in the next section. Because of these properties, they are used for many key functions in current cellular systems, and are likely to be prevalent in future cellular systems as well. In LTE and 5G NR, they are widely used for a number of important initial access and overhead channel functions that are often overlooked by engineers who focus on data transmission. For example, ZC sequences are used for initial access in both the downlink (synchronization sequences) and uplink (random access premables). They are also used for transmitting uplink control information, and as pilot symbols for both uplink channel sounding and fine-grained channel estimation. It is not an exaggeration to say that most types of signals other than the data transmissions in modern cellular standards utilize ZC sequences. In this primer, we define ZC sequences and introduce their key properties, and provide some examples. We also discuss modified ZC sequences that are commonly used in practice, but are not, strictly speaking, ZC sequences. We also overview their uses in LTE and 5G.