We analyze the structure and connectivity of the distinct morphologies that define the Cosmic Web. With the help of our Multiscale Morphology Filter (MMF), we dissect the matter distribution of a cosmological $Lambda$CDM N-body computer simulation into cluster, filaments and walls. The MMF is ideally suited to adress both the anisotropic morphological character of filaments and sheets, as well as the multiscale nature of the hierarchically evolved cosmic matter distribution. The results of our study may be summarized as follows: i).- While all morphologies occupy a roughly well defined range in density, this alone is not sufficient to differentiate between them given their overlap. Environment defined only in terms of density fails to incorporate the intrinsic dynamics of each morphology. This plays an important role in both linear and non linear interactions between haloes. ii).- Most of the mass in the Universe is concentrated in filaments, narrowly followed by clusters. In terms of volume, clusters only represent a minute fraction, and filaments not more than 9%. Walls are relatively inconspicous in terms of mass and volume. iii).- On average, massive clusters are connected to more filaments than low mass clusters. Clusters with $M sim 10^{14}$ M$_{odot}$ h$^{-1}$ have on average two connecting filaments, while clusters with $M geq 10^{15}$ M$_{odot}$ h$^{-1}$ have on average five connecting filaments. iv).- Density profiles indicate that the typical width of filaments is 2$Mpch$. Walls have less well defined boundaries with widths between 5-8 Mpc h$^{-1}$. In their interior, filaments have a power-law density profile with slope ${gamma}approx -1$, corresponding to an isothermal density profile.
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