The characteristic feature of complex systems is the emergence of unexpected properties or behaviour. Complexity, beyond a certain threshold, May even lead to the emergence of new principles. It is a one-way traffic: The new principles and Features may be sometimes deducible from, but are not reducible to, those operating at the lower levels of complexity. Reductionism stands discounted. Complexity is rampant in the animate world. It a rises in in animate systems also, some examples being multiferroic materials and certain nanocomposites, as also a variety of soft-matter systems. Our ecosphere is also a giant, highly complex, open system, which we donot understand much at present. Mastering complexity is the next big challenge for science. Conceptual breakthroughs are needed. In the first part of this article, we introduce the basics of information theory, chaostheory, and computational complexity. In the second part, we shall Describe complex materials as well as some terrestrial and cosmic aspects of complexity.