Magnetic amorphous alloys obtained by rapid quenching of the melt are excellent soft magnetic materials with a wide range of technological applications. They also represent a significant challenge to the scientific understanding of magnetic materials, since most of the existing theories of solids assume lattice periodicity. For these reasons, the magnetic and other properties of amorphous alloys have been very actively studied over the last decade. In recent years increasing attention has been directed to the fundamental understanding of the structural, thermal and magnetic properties of the amorphous alloys. It is not only scientifically but also technologically important to achieve such an understanding, since the amorphous alloys are, in many respects, so different from conventional crystalline magnetic materials. In this review, we attempt to summarise recent progress in research on magnetic amorphous alloys and critically assess the present level of understanding of this new class of magnetic materials, focusing mostly on the transition-metal-metalloid glasses. We start with a review of early developments and a discussion on the nature of glasses and glass formation and proceed to an extensive discussion of their atomic structure, both from the experimental and theoretical points of view. The questions of glass formation, stability and relaxation are then addressed, followed by a review of their non-magnetic properties. The review of magnetic properties starts with a discussion of the magnetic moment, with the emphasis on the relation between electronic structure and magnetic behaviour. The magnetic interaction, local magnetic anisotropy and magnetoelastic effects are then considered, followed by discussions on magnetic excitation and phase transition, magnetic anisotropy and coercivity, and the effect of structural relaxation. Finally we discuss the prospects of technological applications and the manner in which the basic understanding of these properties could benefit such applications.