Mg-based alloys form a variety of interesting structures including stable and metastable crystalline, stable and metastable quasicrystalline, nanocrystalline and amorphous phases. Many of these phases can be made to coexist by suitable processing leading to an interesting combination of properties. Non-equilibrium processing in combination with suitable heat treatments can be used to control the scale and dispersion of these phases. Further thrust to Mg-based alloys is expected through the development of Mg-based bulk metallic glasses. Magnesium matrix composites are also gaining in prominence. The thesis has been divided into theoretical and experimental parts. The theoretical part focuses on understanding the structure of quasicrystals, rational approximants and related structures. The experimental work involves synthesis, non-equilibrium processing and characterization of specific Mg-based alloys. The structure of quasicrystals and related structures can be understood by working in three dimensions or by projection from higher dimensions. The projection formalism is used to generate quasicrystals and rational approximants in 2D and 3D. Approximants to the Penrose lattice are generated with directions of approximation oriented 90° and 72° apart. Rational approximants to the icosahedral quasilattice are generated and the systematics of lattice-centring in these approximants analysed. Two-dimensional quasiperiodic lattice with 5-fold symmetry, which is periodic along the third dimension, is generated as an approximant to the icosahedral lattice. Approximants are also considered wherein quasiperiodicity is retained along one or two directions. The concept of average lattices can be used to understand diverse structures including vacancy ordered phases (VOP) and orthorhombic approximants to the decagonal phase. VOP which lack incommensurate length scales should be considered as quasiperiodic superlattice (QPSL) approximants rather than as conventional rational approximants and hence have the average lattice scheme built into them. The average lattice approach is further used to unify Kuo’s and Anantharaman's models for orthorhombic approximants to the decagonal quasicrystal. A modified version of Anantharaman's model is also presented. Using the twinned icosahedron model, Robinson and Taylor approximants to the decagonal quasicrystal are generated by the twinning of Mackay and Little approximants to the icosahedral quasicrystal. An indexing scheme based on this model is developed which inherits the merits of the twinned icosahedron model. Further, using cluster of four icosahedra, in a distorted tetrahedral configuration, symmetries of the hexagonal phases, which are related to quasicrystals, are generated. Frank's ratio is brought out as a unifying thread connecting diverse kinds of structures including VOP and hexagonal phases related to quasicrystals, which have pseudocubic symmetry. Experimental work involves the synthesis and characterization of alloys in four systems: a) Mg-Zn-Y, b) Mg-Zn-La, c) Al-Mg-Cu and d) Mg-Cu-Y. Induction melting is used to prepare the alloys and melt-spinning is used as the primary non-equilibrium processing route. The focus in the Mg-Zn-RE systems is in the as cast condition while in the Al-Mg-Cu system it is in the melt-spun condition. Characterization techniques used are XRD, SEM and TEM. In the Mg-Zn-Y system face-centred icosahedral (FC1) phase with quasilattice parameter of 5.21 A is found to coexist with related crystalline phases in the Mg4Zn94Y2 and Mg23Zn5gY9 alloys. A series of crystalline phases with superlattice ordering are seen in the Mg-Zn-Y and Mg-Zn-La systems. These phases with a variety of ordering, many of which display interesting patterns of streaking in the SAD pattern, are related to one-another and to the FCI QC found in the Mg-Zn-Y system. No quasicrystal could be observed in the two alloys investigated in the Mg-Zn-La system with La = 5 and 8 %. Conventional rational approximants were conspicuous by there absence in both the rare-earth containing systems. This is understood in terms of the absence of large clusters in these systems. High Y alloys display a tendency to form nanocrystals in the as-cast condition and amorphous regions are observed in the as-cast alloys with Y > 20 %. Hence, high Y alloys are anticipated to be bulk glass formers. Melt-spinning of the alloys in both the RE containing systems lead to the formation of nanocrystalline regions. The e/a ratio plays an important role in the formation of phases in the Mg-Zn-Y system. An e/a ratio near 2.08 has a stabilising effect on a variety of phases including the FCI quasicrystal, ternary phases related to the quasicrystal and binary phases like YZn12 and Y2Zn17. Formation of quasicrystals in the Al-Mg binary and Mg-Al-Cu ternary seem to be very sensitive to processing conditions and were not observed in the present investigation in the melt-spun alloys. However, β-Al3Mg2 and Mg32(Al,Cu)49 phases with large lattice parameters, which are related to quasicrystals, are observed in as-cast and melt-spun conditions. The Mg32(Al,Cu)49 phase brings out the similarity between this system and the Mg-Al-Zn system. The glass formability of the alloys in the Al-Mg binary and in the Mg-Al-Cu ternary is limited. Except for the formation of amorphous phase in some regions, the alloys were crystalline even when melt-spun at 2800 rpm. The ability to form nanocrystals is also limited in this system as compared to the Mg-Zn-RE systems. Often melt-spun alloys showed a wide range of grain sizes coexisting together.