Cracovians are matrices with a redefined product, namely, the row-by-column product is replaced by the column-by-column product (or, alternatively by the row-by-row product). This alteration in the product definition implies that the multiplication of Cracovians is non-commutative and non-associative. An exposition of cracovian algebra on 'Cracovian Calculus', and on the original papers of Banachiewics and his co-workers, together with applications to solutions of Systems of Linear Equations and to Spherical Polygonometry represents the main subjects of this book. The last three sections contain also new applications of cracovian algebra.