This book is a "missing link" between the elementary textbook of Lawvere and Schanuel "Conceptual Mathematics" and the much more advanced textbooks such as the one by MacLane and Moerdijk "Sheaves in Geometry and Logic." Avoiding the complicated, fully fledged notion of a Grothendieck topos, whose very formulation presupposes a good deal of mathematical experience, this book introduces topos theory through presheaf toposes, i.e., readily visualizable categories whose objects result from glueing simpler ones, the "generic figures". Several phenomena which distinguish toposes from the ordinary category of sets appear already at this simpler level. Six easy to understand examples accompany the reader through the whole book, illuminating new material, interpreting general results and suggesting new theorems. This book is aimed (via appropriate examples) at a beginner mathematician or scientist or philosopher who would like to take advantage of the richness of presheaf toposes to prepare himself or herself either for further study or applications of the theory described