This is nearly a century that dimension theory plays a crucial role in topology. Almost zero-dimensionality as a dimension theoretic concept that fits neatly between zero- and one-dimensionality introduced in 1994 and the famous Erdos space is a universal object in the class of all almost zero-dimensional spaces. The main aim of this book is to find the "right" way to extend almost-zero dimensionality, that is, to develop a dimension theory on the basis of this notion. We introduce the notion of an almost n-dimensional space. In this way an important result is a topological analogue of the classic Kadec Renorming Theorem for Banach spaces which produces an elegant topological characterization of our new concept. Construction of higher dimensional analogues of complete Erdos space that are universal spaces for almost n-dimensionality is another interesting result. This book may be readable and useful for anyone interested to mathematical analysis or general topology.