RNA molecules, through their dual appearance as sequence and structure, represent a suitable model to study evolutionary properties of quasispecies. The essential ingredient in this model is the differentiation between genotype (molecular sequences, affected by mutation) and phenotype (molecular structure, affected by selection). This framework allows a quantitative analysis of organizational properties of quasispecies as they adapt to different environments, such as their robustness, the effect of the degeneration of the sequence space, or the adaptation under different mutation rates and the error threshold associated. First, we investigate the mapping between sequence and structure spaces in RNA populations. Analysis of natural RNAs show that certain structural motifs are much more abundant than others, pointing out a complex relation between sequence and structure. We have investigated computationally the structural properties of a large pool (10E8 molecules) of single-stranded, 35 nt-long, random RNA sequences. The secondary structures obtained are ranked and classified into structure families. The number of structures in main families is analytically calculated and compared with the numerical results. This permits a quantification of the fraction of structure space covered by a large pool of sequences. We further show that the number of structural motifs and their frequency is highly unbalanced with respect to the nucleotide composition. Then, we describe and analyse the structural properties of molecular quasispecies adapting to different environments both during the transient time before adaptation and in the asymptotic state. We observe a minimum in the adaptation time at values of the mutation rate relatively far from the error threshold. Through the definition of a consensus structure, it is shown that the quasispecies retains relevant structural information in a distributed fashion even above the error threshold. This structural robustness depends on the precise shape of the secondary structure used as target of selection.