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We introduce the notion of a braided algebra and study some examples ofthese. In particular, R-symmetric and R-skew-symmetric algebras of a linearspace V equipped with a skew-invertible Hecke symmetry R are braided algebras.We prove the "mountain property" for the numerators and denominators of theirPoincare-Hilbert series (which are always rational functions).  Also, we further develop a differential calculus on modified ReflectionEquation algebras. Thus, we exhibit a new form of the Leibniz rule for partialderivatives on such algebras related to involutive symmetries R. In particular,we present this rule for the algebra U(gl(m)). The case of the algebra U(gl(2))and its compact form U(u(2)) (which can be treated as a deformation of theMinkowski space algebra) is considered in detail. On the algebra U(u(2)) weintroduce the notion of the quantum radius, which is a deformation of the usualradius, and compute the action of rotationally invariant operators and inparticular of the Laplace operator. This enables us to define analogs of theLaplace-Beltrami operators corresponding to certain Schwarzschild-type metricsand to compute their actions on the algebra U(u(2)) and its central extension.Some "physical" consequences of our considerations are presented.

Braided algebras and their applications to Noncommutative Geometry