Non-abelian topological states of matter are of great interest in condensed matter physics,both due to their extraordinary fundamental properties and to their possible use for quantumcomputation. The insensitivity of their topological characteristics to disorder, noise,and interaction with the environment may lead to realization of quantum computers withvery long coherence times. The realization of a quantum computer ranks among the foremostoutstanding problems in physics, particularly in light of the revolutionary rewardsthe achievement of this goal promises.The proposed theoretical study is multi-dimensional. On the methodological side themulti-dimensionality is in the breadth of the studies we discuss, ranging all the way fromphenomenology to mathematical physics. We will aim at detailed understanding of presentand future experimental results. We will analyze experimental setups designed to identify,characterize and manipulate non-abelian states. And we will propose and classify novelnon-abelian states. On the concrete side, the multi-dimensionality is literal. The systemswe consider include quantum dots, one dimensional quantum wires, two dimensional planarsystems, and surfaces of three dimensional systems.Our proposal starts with Majorana fermions in systems where spin-orbit coupling, Zeemanfields and proximity coupling to superconductivity are at play. It continues with “edgeanyons”, non-abelian quasiparticles residing on edges of abelian Quantum Hall states. Itends with open issues in the physics of the Quantum Hall Effect.We expect that this study will result in clear schemes for unquestionable experimentalidentification of Majorana fermions, new predictions for more of their measurable consequences,understanding of the feasibility of fractionalized phases in quantum wires, feasibleexperimental schemes for realizing and observing edge anyons, steps towards their classification,and better understanding of quantum Hall interferometry.