Direct-Sequence Code Division Multiple Access (DS-CDMA), over the last few years, has become a popular technique and finds a place in many modern communication systems. The performance of this technique is closely linked to the signature (or spreading) sequences employed in the system. In the past, there have been many successful attempts by research groups to construct families of signature sequences that offer the potential gains promised by theoretical bounds. In this thesis, we present constructions of families of signature sequences over the AM-PSK and QAM alphabet with low correlation. In this thesis, we construct a family of sequences over the 8-ary AM-PSK constella- tion, Family AOpt(16) that is asymptotically optimal with respect to the Welch bound on maximum magnitude of correlation for complex sequences. The maximum magnitude of correlation for this family, θmax, is upper bounded by √N , where N is the period of the sequences. The 8-ary AM-PSK constellation is a subset of the 16-QAM constellation. We also construct two families of sequences over 16-QAM, Family A16A, and Family A16,B , with the maximum magnitude of correlation upper bounded by √2√N . We construct a family, A(M 2), of sequences over the 2m+1-ary AM-PSK constellation of period N = 2r- 1 and family size (N + 1)/2m-1 . The 2m+1-ary AM-PSK constellation is a subset of the M 2-QAM constellation with M =2m . The maximum nontrivial normalized correlation parameter is bounded above by θmax < a √N where a ranges from 1.34 in the case of M 2 = 16 to √5 for large m. Apart from low correlation values, the family possesses several interesting and useful features. In Family A(M 2), users have the ability to transmit 2m bits of data per period of the spreading sequence. The sequences in Family A(M 2) are balanced; all points from the 2m+1-ary AM-PSK constellation occur approximately equally often in sequences of long period. The Euclidean distance between the signals assigned to a particular user in A(M 2), corresponding to different data symbols, is larger than the corresponding value for the case when 2m+1-PSK modulation and spreading is used. Perhaps most interestingly, Family A(M 2) permits users on the reverse link of a CDMA system to communicate asynchronously at varying data rates by switching between different QAM constellations. Family A(M 2) is compatible with QPSK sequence families S(p) in the sense that the maximum correlation magnitude is increased only slightly if one adds sequences from (p) S(p)\ S(0) to Family A(M 2). We also construct families of sequences over AM-PSK that tradeoff data rate per sequence period and θmax for a given family size. We have extended the construction of sequences over AM-PSK constellation to construct sequences over the M 2-QAM constellation for M =2m . The QAM sequence families, Families (AM 2), have size, data rate and minimum squared Euclidean distance same as the corresponding AM-PSK construction but have higher values of θmax. Also included in the thesis are constructions for large families of sequences over the M 2-QAM alphabet.