  This work is intended as a pedagogical introduction to M-theory and to its
maximally supersymmetric toroidal compactifications, in the frameworks of 11D
supergravity, type II string theory and M(atrix) theory. U-duality is used as
the main tool and guideline in uncovering the spectrum of BPS states. We review
the 11D supergravity algebra and elementary 1/2-BPS solutions, discuss
T-duality in the perturbative and non-perturbative sectors from an algebraic
point of view, and apply the same tools to the analysis of U-duality at the
level of the effective action and the BPS spectrum, with a particular emphasis
on Weyl and Borel generators. We derive the U-duality multiplets of BPS
particles and strings, U-duality invariant mass formulae for 1/2- and 1/4-BPS
states for general toroidal compactifications on skew tori with gauge
backgrounds, and U-duality multiplets of constraints for states to preserve a
given fraction of supersymmetry. A number of mysterious states are encountered
in D<=3, whose existence is implied by T-duality and 11D Lorentz invariance. We
then move to the M(atrix) theory point of view, give an introduction to
Discrete Light Cone Quantization (DLCQ) in general and DLCQ of M-theory in
particular. We discuss the realization of U-duality as electric-magnetic
dualities of the Matrix gauge theory, display the Matrix gauge theory BPS
spectrum in detail, and discuss the conjectured extended U-duality group in
this scheme.
