The acquisition of biological data, ranging from molecular characterization and simulations (e.g. protein folding dynamics), to systems biology endeavors (e.g. whole organ simulations) all the way up to ecological observations (e.g. as to ascertain climate changes impact on the biota) is growing at unprecedented speed. The use of computational and networking resources is thus unavoidable. As the datasets become bigger and the acquisition technology more refined, the biologist is empowered to ask deeper and more complex questions. These, in turn, drive a runoff effect where large research consortia emerge that span beyond organizations and national boundaries. Thus the need for reliable, robust, certified, curated, accessible, secure and timely data processing and management becomes entrenched within, and crucial to, 21st century biology. Furthermore, the proliferation of biotechnologies and advances in biological sciences has produced a strong drive for new informatics solutions, both at the basic science and technological levels. The previously unknown situation of dealing with, on one hand, (potentially) exabytes of data, much of which is noisy, has large experimental errors or theoretical uncertainties associated with it, or on the other hand, large quantities of data that require automated computationally intense analysis and processing, have produced important innovations in web and grid technology. In this paper we present a trace of these technological changes in Web and Grid technology, including details of emerging infrastructures, standards, languages and tools, as they apply to bioinformatics, computational biology and systems biology. A major focus of this technological review is to collate up-to-date information regarding the design and implementation of various bioinformatics Webs, Grids, Web-based grids or Grid-based webs in terms of their infrastructure, standards, protocols, services, applications and other tools. This review, besides surveying the current state-of-the-art, will also provide a road map for future research and open questions.