Understanding how materials fracture is a fundamental problem of science and engineering even today, although the effects of geometry, loading conditions, and material characteristics on the strength of materials have been investigated since ancient times. Disorder and long-range interactions are two of the key components that make material failure an attractive subject for the application of statistical mechanics. The cornerstone in this respect has been lattice models of the fracture in which networks with prescribed bond failure thresholds are subject to increasing external loads. These models describe on a qualitative level the failure processes of real, brittle or quasi-brittle materials. This has been particularly important in solving the classical engineering problems of material strength: the size-dependence of maximum stress and its sample-to-sample statistical fluctuations. At the same time, lattice models pose many fundamental questions in statistical physics, such as the relation between fracture and phase transitions. Experimental results reveal the existence of an intriguing crackling noise in the acoustic emission and of self-affine fractals in the crack surface morphology. Recent advances in computer power have enabled considerable progress in the understanding of such models. Among these still partly controversial issues, are the scaling and size effects in material strength and accumulated damage, the statistics of avalanches or bursts of microfailures, and the morphology of the crack surface. Here, using large scale numerical simulations and extensive statistical sampling, we present the statistical properties of fracture in two- and three-dimensional discrete lattice models, in particular, the random fuse model. For two-dimensional systems, we considered lattice systems of sizes up to L = 1024, and for three-dimensional systems, we considered systems of sizes up to L = 64, which are the largest ever systems analyzed for obtaining the scaling laws of fracture. In addition, in two-dimensions, we also considered notched samples of sizes up to L = 512 with varying degrees of disorder. Based on these simulations, we analyze the effect of disorder and the presence of notch on fracture strength distribution, the size-effect on the mean fracture strength, and finally the multi-scaling of crack surfaces and its roughness.