The technological foundations and businesses models upon which digital companies currently function, the so-called ‘platforms’, are characterized by the sort of features that attract antitrust scrutiny. They are usually conformed as multi-sided markets; they pervasively exhibit network effects; the nature and scope of entry barriers is uncertain; dominance in each market is duly granted to the successful company; foreclosure of small firms is always wandering around. Thus, it’s no surprise that digital markets champions such as Facebook, Amazon, Google and Apple have received close attention from the regulators in both sides of the Atlantic. Accordingly, antitrust enforcement efforts both in Europe and the United States have recently increased in number of actions taken and the amount of fines imposed in high-tech industries. Recent decisions in this field have rekindled in-depth and largely unresolved debates concerning the appropriate role of antitrust enforcement in such markets. Nevertheless, the Statement of Objections sent recently by the European Commission to Google, for allegedly abusing its dominant position in the internet-search market goes a step further, jeopardizing the very goals and objectives competition policy is supposed to pursue. Against this background and in light of other recent high-profile cases (Intel, Tomra, Post Danmark, and the everlasting Microsoft) this paper seeks to set an adequate legal and economic framework for the prohibition of monopolization under article 102 of the Treaty of Functioning of the European Union (TFUE). This sort of high tech markets are usually characterised for a heavy reliance in ICT (information and communication technologies), the importance of product design and the development of platforms. In addition, on the basis of ‘winner-takes-it-all’ paradigm, the platform on which they operate may be considered as an ‘essential facility’. As a result, any discrimination or denial of access may be considered abusive.However, the legal treatment to be applied to ‘refusals to supply’ remains as one of the most contentious issues in contemporary EU competition law. It was at the heart of Microsoft’s misbehaviour in 2004 -- and all the subsequent fines that followed -- and, ten years later, if Google’s conduct regarding discrimination of competing firms in favour of its own services may walk the very same path. The four topics addressed in this paper concern three long-lasting controversies and one recent matter: i) under what circumstances a dominant firm is entitled to respond -- albeit, aggressively -- to its competitor’s challenges; ii) the precise scope -- if there’s any -- of the ‘objective justification’ defence; iii) if that sort of conduct is prohibited only when leads to anticompetitive foreclosure, without restrictive effects, is there room for a ‘by object’ interdiction of abuse of a dominant position?; iv) finally, in the specific digital industry, and given the still uncertain economics of multi-sided market and the evolving nature of prevalent businesses models, is really a ‘monopoly’ as bad and harmful as with the bricks-and-mortar or oil-well-and-pipeline predecessors? The ultimate question is, does current antitrust enforcement in these specific markets achieve its goal of fostering competition or rather the undesired effect of chilling innovation?