Miller Urey Experiment
Certainly the most famous, this experiment was proposed by Urey in the 1950's, then demonstrated
by Miller.
They recreated a pre-biotic Earth in the lab, assuming a highly reducing environment (no O2, lots of NH3 and CH4), liquid water heated to get a vapor, then sparking their primitive atmosphere (simulating natural energy sources such as lightning, UV radiation or cosmic rays) and letting the result condense back to liquid water. This was left for a few weeks, then the water in the condensing column was collected and analyzed. Indeed, about 1/6th the C had formed into complex organic molecules, including amino acids!
What's wrong? This works only for a very reducing atmosphere, predominant with NH3 and CH4, which is no longer thought to be correct.
Hydrothermal Vents on the Ocean Floor. Water percolated through the rock contains highly oxidizing molecules, which provide energy for autotrophic microorganisms to form organic compounds. Recall genetic evidence that early life was hypothermophilic, conditions of early earth (mostly oceanic crust), and the safety of deep oceans from large meteoritic impacts. This is now a favorite theory.
Incoming Meteorites. Comets and asteroids contain rather complex organic molecules (including amino acids) some of which should be able to make it through the atmosphere and land on Earth's surface. Also, the shock wave of their entry to Earth's atmosphere may have produced enough energy for complex molecule formation.