This subject is on the border of Number Theory, Geometry, Combinatorics, and Probability Theory. The basic question is obvious: given a ``nice" region in the plane, what can one say about the number of lattice points in the region? Of course, there are ``easy" cases and ``hard" cases. The well-known Pick's theorem is an ``easy" case: if the region is a lattice polygon (each vertex is a lattice point), then there is a simple relation between the area, the number of points on the border, and inside. The classical Circle Problem (due to Gauss) is an example of a ``hard" question: if the region is a circle centered at the origin, then the number of lattice points is very close to the area, and as the radius changes, one expects fluctuations of size about the ``square-root of the circumference". This old conjecture is wide open, in spite of all efforts by the best experts in analytic number theory in the last 100 years. Is there a ``natural" region for which we can still prove what one would expect? The answer is yes; these natural regions are (1) long, narrow hyperbola-segments; (2) tilted squares; (3) triangles, and so on. We can prove perfect ``deterministic" analogs of well-known probabilistic results like the Law of the iterated logarithm and the Central limit theorem.