Within the scientific research community, memory information in the brain is commonly believed to be stored in the synapse - a hypothesis famously attributed to psychologist Donald Hebb. However, there is a growing minority who postulate that memory is stored inside the neuron at the molecular (RNA or DNA) level - an alternative postulation known as the cell-intrinsic hypothesis, coined by psychologist Randy Gallistel. In this paper, we review a selection of key experimental evidence from both sides of the argument. We begin with Eric Kandel's studies on sea slugs, which provided the first evidence in support of the synaptic hypothesis. Next, we touch on experiments in mice by John O'Keefe (declarative memory and the hippocampus) and Joseph LeDoux (procedural fear memory and the amygdala). Then, we introduce the synapse as the basic building block of today's artificial intelligence neural networks. After that, we describe David Glanzman's study on dissociating memory storage and synaptic change in sea slugs, and Susumu Tonegawa's experiment on reactivating retrograde amnesia in mice using laser. From there, we highlight Germund Hesslow's experiment on conditioned pauses in ferrets, and Beatrice Gelber's experiment on conditioning in single-celled organisms without synapses (Paramecium aurelia). This is followed by a description of David Glanzman's experiment on transplanting memory between sea slugs using RNA. Finally, we provide an overview of Brian Dias and Kerry Ressler's experiment on DNA transfer of fear in mice from parents to offspring. We conclude with some potential implications for the wider field of psychology.