In 2009, in Lawrence Livermore National Laboratory, USA, National Ignition Facility (NIF) - the largest thermonuclear fusion device ever made was launched. Its main part is a multi-beam laser whose energy in nanosecond pulse exceeds 1MJ (106 J). Its task is to compress DT fuel to the density over a few thousand times higher than that of solid-state DT and heat it to 100 millions of K degrees. In this case, the process of fuel compression and heating is realized in an indirect way - laser radiation (in UV range) is converted in the so-called hohlraum (1 cm cylinder with a spherical DT pellet inside) into very intense soft X radiation symmetrically illuminating DT pellet. For the first time ever, the fusion device's energetic parameters are sufficient for the achieving the ignition and self-sustained burn of thermonuclear fuel on a scale allowing for the generation of energy far bigger than that delivered to the fuel. The main purpose of the current experimental campaign on NIF is bringing about, within the next two-three years, a controlled thermonuclear 'big bang' in which the fusion energy will exceed the energy delivered by the laser at least ten times. The expected 'big bang' would be the culmination of fifty years of international efforts aiming at demonstrating both physical and technical feasibility of generating, in a controlled way, the energy from nuclear fusion in inertial confined plasma and would pave the way for practical realization of the laser-driven thermonuclear reactor. This paper briefly reviews the basic current concepts of laser fusion and main problems and challenges facing the research community dealing with this field. In particular, the conventional, central hot spot ignition approach to laser fusion is discussed together with the more recent ones - fast ignition, shock ignition and impact ignition fusion. The research projects directed towards building an experimental laser-driven thermonuclear reactor are presented as well.