Continued development of relatively compact, although not quite 'table-top', lasers with peak powers in the range up to 100 TW has enabled laser-plasma-based acceleration experiments with amazing gradients of up to 1 TV/m. In order to usefully apply such gradients to 'controlled' acceleration, various hurdles need to be overcome. The main one is that of well-synchronized injection into a sub-mm to micron wavelength plasma wave. The talk will describe the various physics regimes of laser wakefield acceleration, and the two classes of experiments being pursued. One is that of atmospheric-density plasmas, non-linear wakefields with extreme gradients (hundreds of GV/m)and 'internal injection' of few-femtosecond electron bunches. A second class ...