The best and most complete theory of transport in magnetically confined systems is the Neoclassical theory. However, it is found that transport often exceeds Neoclassical expectations by an order of magnitude or more (also see Non-diffusive transport).  The difference between actual transport and the Neoclassical expectation is called "anomalous" transport. It is generally assumed that the anomalous component of transport is generated by turbulence driven by micro-instabilities. 
How important is turbulence?
In spite of lengthy studies into the subject, it is still controversial how important turbulent transport really is. In part, this may be because turbulent transport gives a variable contribution to transport (depending on local and global parameters), whereas Neoclassical transport is always present. And in part, because no complete theory for anomalous transport is available. 
An important argument suggesting that anomalous transport is important to the degree that it often dominates the total transport is the scaling of transport with heating power and machine size.  The phenomenon of power degradation, universally observed in all devices, is an indication that standard transport theories are inadequate to explain all transport, since these would not predict power degradation. Following Freidberg,  the cited scaling laws can be rewritten in terms of the temperature dependence (eliminating the heating power dependence). Then, classical and neoclassical estimates would predict that the confinement increases with T (namely: τE ∝ T0.5, associated with collisionality). However, the experimental scalings give a decrease with T (namely: τE ∝ Tα with α < -1). This unexpected behaviour is explained from increased turbulence levels (and enhanced transport) at higher values of (the gradients of) T.
Profile consistency indicates that self-organisation plays an important role in transport, and this can only be the case when instabilities or turbulence are able to regulate the profiles, i.e., when they carry an important fraction of transport.
The suppression of turbulence is possible, either actively (by imposing an external radial electric field), or spontaneously (H-modes, Internal Transport Barriers). As a consequence, transport is reduced significantly (to Neoclassical levels). This is a clear indication that turbulence is responsible for the main fraction of anomalous transport.
It has been argued that turbulence cannot be responsible for a significant fraction of the anomalous component of transport, since that would lead to high resistivity (due to collisions), which contradicts experimental observation.  However, this argument fails to note that anomalous transport may consist of collective events (e.g., streamers), which does not require an enhanced collisionality. As a side remark, this argument does show that the contribution of turbulence to transport is likely not of the diffusive type (see Non-diffusive transport).
The physical mechanism behind anomalous transport has not been fully clarified. However, it is generally assumed that anomalous transport is the consequence of microscopic instabilities. The plasma potentially produces a plethora of such instabilities, due to the fact that it is in a state far from thermodynamic equilibrium, with steep density, temperature, and pressure gradients. The most likely candidates involved in generating the observed anomalous transport are: 
- Ion Temperature Gradient (ITG) instabilities
- Electron Temperature Gradient (ETG) instabilities
- Collisionless Trapped Electron Modes (TEM)  
- Dissipative Trapped Electron Modes (DTEM)
(to be completed; references needed)
Can anomalous transport be modelled?
There are several answers to this question. Since all equations describing the motion of charged particles in fields are known, as well as the effects of collisions, detailed numerical (gyrokinetic) simulations are possible.  However, due to the enormous disparity between the minimum and maximum scales involved (gyration times vs. transport times, and the gyroradius vs. the machine size), this is a major challenge.
An alternative approach is to model the net effect of turbulence without simulating the fine detail. In doing so, it is not sufficient to introduce a simple additional "turbulent diffusivity", as this cannot possibly reproduce the observed global transport scaling behaviour. It is probably necessary to use a non-diffusive description,  and include non-linear phenomena such as critical gradients.
Can anomalous transport be controlled?
Yes. The impression is that anomalous transport is more difficult to control in tokamaks than in stellarators. However, limited control in tokamaks is possible by making use of edge transport barriers (cf. H-mode) and Internal Transport Barriers (ITBs). This reduces transport to Neoclassical levels, at least transiently and locally.
Particularly in optimised stellarators (W7-AS), transport can be close to Neoclassical levels. 
- A.J.Wootton et al, Fluctuations and anomalous transport in tokamaks, Phys. Fluids B 2 (1990) 2879
- J.P. Freidberg, Plasma physics and fusion energy, Cambridge University Press (2007) ISBN 0521851076
- J.W. Conner and H.R. Wilson, Survey of theories of anomalous transport, Plasma Phys. Control. Fusion 36 (1994) 719-795
- B.A. Carreras, Progress in anomalous transport research in toroidal magnetic confinement devices, IEEE Trans. Plasma Science 25, 1281 (1997)
- L.C. Woods, Theory of tokamak transport: new aspects for nuclear fusion reactor design, John Wiley and Sons (2006) ISBN 3527406255
- J. Weiland, Collective modes in inhomogeneous plasma: kinetic and advanced fluid theory, Plasma physics series, CRC Press (2000) ISBN 0750305894
- B. Coppi and G. Rewoldt, New Trapped-Electron Instability, Phys. Rev. Lett. 33 (1974) 1329 - 1332
- F. Ryter et al, Experimental Study of Trapped-Electron-Mode Properties in Tokamaks: Threshold and Stabilization by Collisions, Phys. Rev. Lett. 95 (2005) 085001
- A.M. Dimits et al, Scalings of Ion-Temperature-Gradient-Driven Anomalous Transport in Tokamaks, Phys. Rev. Lett. 77 (1996) 71 - 74
- G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports 371, Issue 6 (2002) 461-580
- M. Hirsch et al, Major results from the stellarator Wendelstein 7-AS, Plasma Phys. Control. Fusion 50 (2008) 053001