The plot of this film initially seems like a standard "accidental superhero" story: heroine Lucy (Scarlett Johansson) receives a massive overdose of a new drug and acquires superpowers which enable her to gain revenge on those who have mistreated her. In reality, however, that is only the starting point of a highly ambitious tale which heads off in an unusual direction. The problem is that the basic premise of the story is flawed but, if you can swallow that, the film is well worth watching. If you'd rather find out for yourself then you had better stop reading as the rest of this review contains spoilers.
Let's get the basic flaw out of the way first: that is the claim that humans only use a small percentage of our brains and that anything would be possible if we could substantially increase this. We see a scientist (Morgan Freeman) giving a lecture on this subject in a parallel plot thread which converges with the main thread when Lucy tracks him down to ask for his help. The problem is that this widely-popular notion is not true; neuroscientists, who have ever more sophisticated tools for studying how our brains work, state that we do in fact have a use for every part of them. This should come as no surprise, given the evolutionary imperative of "use it or lose it"; our huge brains absorb huge resources to make and maintain, and if we failed to use 90% of them, we wouldn't keep them for long.
This rather undermines the whole basis of the plot, and the dramatic way in which a number flashes up on screen from time to time to show the steady percentage increase in Lucy's use of her brain as the drug increases the neural connections. Also unconvincing are the consequences of this in terms of instant superpowers, fully developed and under control. Ultimately she attains near god-like powers, but she realises that this rapid evolution of her cells will have an inevitable conclusion within a matter of hours: her own death. There are echoes here of Flowers for Algernon, although Lucy is much more complicated, messy, fast-moving and violent than Keyes' classic tale.
Despite the flaws there is much to enjoy. First and foremost is Johansson's performance; I am not a fan of hers but I have to admit that she carries this film, giving an excellent performance while appearing in almost every scene. Then there are the playful moments that Luc Besson inserts: initially, as the helpless innocent Lucy has gang members closing in on her, the action is interspersed with clips of predators menacingly closing in on their prey. The impressive CGI takes centre stage in the finale, and the whole film is packed with such interesting detail that I would happily sit down and watch it again – not something I could say about many films.