Comets are probably the best archives of the nascent solar system, 4.5 Gyr ago, and their compositions reveal crucial clues on the structure and dynamics of the early protoplanetary disk. Anhydrous minerals (olivine and pyroxene) have been identified in cometary dust for a few decades. Surprisingly, samples from comet Wild2 returned by the Stardust mission in 2006 also contain high temperature mineral assemblages like chondrules and refractory inclusions, which are typical components of primitive meteorites (carbonaceous chondrites - CCs). A few Stardust samples have also preserved some organic matter of comet Wild 2 that share some similarities with CCs. Interplanetary dust falling on Earth originate from comets and asteroids in proportions to be further constrained. These cosmic dust particles mostly show similarities with CCs, which in turn only represent a few percent of meteorites recovered on Earth. At least two (rare) families of cosmic dust particles have shown strong evidences for a cometary origin: the chondritic porous interplanetary dust particles (CP-IDPs) collected in the terrestrial stratosphere by NASA, and the ultracarbonaceous Antarctic Micrometeorites (UCAMMs) collected from polar snow and ice by French and Japanese teams. Analyses of dust particles from the Jupiter family comet 67P/Churyumov-Gerasimenko by the dust analyzers on Rosetta orbiter (COSIMA, GIADA, MIDAS) suggest a relationship to interplanetary dust/micrometeorites. A growing number of evidences highlights the existence of a continuum between asteroids and comets, already in the early history of the solar system.