Qualitatively proposed by Mariotte and mathematically formulated by Weibull, the statistical theory of size effect reigned supreme until the 1980s. However, beginning in 1976, a different, non statistical theory of size effect gradually emerged and was shown to be important for brittle heterogeneous materials, concisely termed "quasibrittle", which include concrete, fiber composites, sea ice, rocks, stiff cohesive soils, snow slabs, foams, wood, paper, bone and many materials on micrometer scale. In these materials, the maximum load is attained only after the stable formation of either (1) a large fracture process zone (FPZ), with distributed cracking, or (2) a large crack. The lecture first reviews the sources of these type I and II size effects, consisting of the energy release associated with stress redistribution prior to maximum load, and combined in Type I size effect with the statistical weakest link model for a finite chain. Numerical simulation, requiring nonlocal and stochastic approaches, is discussed. The asymptotic matching approach (anchored through dimensional analysis) to the development of analytical laws for Type I and Type II quasibrittle size effects, is then outlined. It is explained that while in Type 2 material randomness affects only the variance and the tail of the cumulative distribution of RVE strength is unimportant, in Type I it also strongly (though not totally) affects the mean, and the far-left tail plays a major role. This role can be theoretically predicted by nano-fracture mechanics of cracks in the atomic lattice based on crack length dependence of the activation energy barriers for the metastable states of the free energy potential of the atomic lattice. Extensions to reentrant corners, time dependence, and cyclic loading are pointed out. Various structural applications, experimental evidence, and reinterpretation of some structural disasters are discussed. Finally, some important open problems are highlighted.