Summary: A new class of \(q\)-ary sequences, called interleaved sequences, is introduced. Their periods, shift equivalence, linear spans, and autocorrelation functions are derived. The interleaved sequences include a large number of popular sequences, such as multiplexed sequences, clock-controlled sequences, Kasami sequences, GMW sequences, geometric sequences, and No sequences. A special class of the interleaved sequences is constructed by mapping \(\text{GF}(q^m)\) sequences into \(\text{GF}(q)\) sequences in terms of different bases of \(\text{GF}(q^m)\) over \(\text{GF}(q)\). As an application of the theory of interleaved sequences, some new families of binary pseudo-random sequences are constructed, which have large linear spans, optimal periodic cross/autocorrelation functions, balance, and the rapidly `hopped' properties. A complete comparison of the new family of sequences with the Gold sequence family, the Kasami (small and large set) sequence families, the bent-function sequence family, and the No sequence family is discussed. This shows that the new sequence families have important advantages for use in spread-spectrum multiple-access communication systems.