The celebrated Curvature-Dimension condition CD(K,N) of Bakry and Emery is a well-known generator of isoperimetric, functional and concentration inequalities. Traditionally, the range of admissible values for the generalized dimension N has been confined to [n,\infty], where n is the (topological) dimension of the ambient space. We extend this in two manners. First, we treat the range N < 1, allowing in particular negative generalized dimensions, and identifying new one-dimensional model-spaces for the isoperimetric problem. Of particular interest is when curvature is strictly positive, yielding a new single model space (besides the previously known N-sphere and Gaussian measure): a positively curved two-sided hyperbolic space of (possibly negative) dimension N < 1, enjoying a two-level concentration a-la Talagrand. When curvature is only assumed non-negative, we confirm that such spaces always satisfy an N-dimensional Cheeger isoperimetric inequality and N-degree polynomial concentration, and establish that these properties are in fact equivalent. In particular, this renders equivalent various weak Sobolev and Nash inequalities for different exponents on such spaces. Second (time permitting), to treat the forbidden range 1 < = N < n, where all the well-understood isoperimetric, Sobolev and concentration properties completely break-down for the traditional CD condition, we propose a novel graded extension of the CD condition. The Graded CD condition reflects a scenario in which the space decouples into two metric-measure "halves" contributing to the total Curvature-Dimension, such that each half satisfies a curvature lower bound requirement. As our main application, we will present (time permitting) improved isoperimetric inequalities in Euclidean space.