This book is a study of fixed point theory and the underlying spaces, viz. Metric, Symmetric, D_-metric and Topological vector space valued cone spaces. After a comprehensive general introduction to the terminologies of fixed point theory, the present book abounds theorems with proofs in considerable detail, examples and counter-examples. It aims to delineate a comparative study of various types of compatible maps, including weakly compatible maps in metric Spaces. It attempts to characterize occasionally weakly compatible maps with weakly biased maps.