This is a comprehensive and state-of-the-art compendium of Newtonian-Eulerian (momentum/vectorial) mechanics, from an advanced and unified (i.e. continuum) viewpoint — “elementary” simply means no Lagrangean and no Hamiltonian theories / methods, otherwise the level is graduate and beyond; yet, the presentation is eminently readable and inclusive: the style is informal and the mathematics (this Achilles heel of modern mechanics books, especially of the applied/multi-body ones) has been kept to the simplest necessary for a work of this level and scope (see below) — all ahistorical and intuition-deadening jargon and formalisms have been intentionally avoided. As such, the book continues, renovates, and expands the grand tradition laid by such educational 20th century “elementary mechanics” classics as those written by (alphabetically): Butenin, Coe, Fox, Hamel, Lur’e, Magnus, Milne, Nielsen, Parkus, Pars, Synge, Szabó, Voigt et al.Specifically: The book opens with a unique part — “sub-book” (“zeroth approximation”) — on the history of theoretical and applied mechanics — from classical antiquity (Greece) to late 20th cent., including many hundreds of portraits and photos of famous researchers in mechanics, and famous quotes; Next — and to make this volume as self-contained as possible, and establish better communication with the readers — an extensive, handbook-like, exposition of the necessary mathematics is presented, i.e. algebra + calculus of Cartesian vectors and tensors, and linear algebra / matrices (in both indicial and direct notations); Then, the main course follows, i.e. a detailed treatment of the fundamentals of (the exact, nonlinear) kinematics, kinetics (and some thermodynamics) of particles, rigid bodies, and deformable continua (i.e. of Euler, Cauchy, Hamel et al.); most basic concepts and general equations are discussed, as well as their approximations (linearizations etc.) — the emphasis is on ideas and results of permanent interest.The text is complemented by many remarks and completely solved nontrivial examples, along with many problems (all with their answers and many with hints).