For the first time, Swift is giving us the opportunity to study supergiant fast X-ray transients (SFXTs) throughout all phases of their life: outbursts, intermediate level, and quiescence. We present our intense monitoring of four SFXTs, observed 2-3 times per week since October 2007. We find that, unexpectedly, SFXTs spend most of their time in an intermediate level of accretion ($L_{X}sim 10^{33-34} $ erg s$^{-1}$), characterized by rich flaring activity. We present an overview of our investigation on SFXTs with Swift, the key results of our Project. We highlight the unique contribution Swift is giving to this field, both in terms of outburst observations and through a systematic monitoring.