We introduce a class of multifractal processes, referred to as multifractal random walks (MRWs). To our knowledge, it is the first multifractal process with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio. The MRWs are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures.