Recently knowledge-based methods for the in-silico prediction of protein structure reached a considerable success, in part due to the growing wealth of experimentally resolved protein structures. In the future these methods may further replace expensive structural biology experiments leading to the efficient exploitation of genomes in drug discovery. Physics-based methods for the de-novo prediction of the native state are instead still very expensive: e.g. all-atom simulations are applied just to peptides and few small proteins. However, the latter approaches have the major advantage of providing atomic-scale information on the dynamics of the folding process [1-3].
In the last years, the timescales reachable by accurate simulations enlarged considerably (thanks e.g. to distributed computing , enhanced sampling techniques [5-8], specialized hardware , etc.), while those probed by experiments reduced (fast probes [10-14], synthesis of fast folding peptides [15-16], etc.). Nowadays the two timescales start to overlap sizably, with a consequent great advance in the comprehension of folding mechanisms [16-20]. It emerges that proteins can fold through a complicated network of pathways including several intermediates on- or off-pathway [21-23]. Unfolded and intermediate states are often characterized by structural heterogeneity. The prevailing pathway depends on conditions like pH, denaturants, temperature, and is affected also by mutations or ligand binding. Even when simple exponential kinetics is observed, it does not necessarily imply a simple one-route two-state folding pathway . Nanoscale dewetting (water drying) can also play a significant role in the protein folding kinetics. 
Simulations can help answering the relevant open questions: How the amino acid sequence determines the native structure and folding pathways? In which precise way the structural heterogeneity reduces during folding (i.e., what is the shape of the folding funnel)? What are the physical driving forces of folding? To which extent the existing simplified models of folding mechanisms are realistic? To answer these questions, simulations face challenges: How to sample efficiently the space of protein conformations? How to extract relevant information from the huge amount of data stored in atomic trajectories? How to compare the complex kinetic schemes emerging from simulations with the limited information obtained from experiments? An emerging paradigm for the interpretation of protein folding dynamics is that of Markovian master equations [4, 8, 24]. These approaches provide a coarse-grained visualization of folding pathways as a sequence of transitions among intermediate states, and in some cases they can be validated by the most advanced experimental probes. Thanks to the combined use of theories, experiments, and simulations, the features of the free-energy landscapes through which proteins fold are getting clearer: insight is emerging about the effective size and heterogeneity of the conformational space, the nature of the unfolded ensemble, and the reaction coordinates which drive folding [1, 25-33].