The Finite Cell Method (FCM) is highly efficient for simulating geometrically complex solids by embedding the physical domain inside a simple structured mesh and assigning material properties at integration points. However, classical FCM inherits the continuity constraints of standard finite elements, making it less suitable for problems involving cracks, slits, or multi-material interfaces. In this setting, a strong discontinuity refers to a jump in the primary field, such as displacement across a crack, while a weak discontinuity means the field remains continuous but its gradient, such as strain or stress, exhibits a jump. The Discontinuous Finite Cell Method (DFCM) extends FCM by removing inter-element continuity constraints and introducing variational coupling through Nitsche’s method and numerical fluxes. Each finite cell acts as an independent subdomain, and continuity is enforced weakly through consistent interface terms. This allows the method to represent geometric and material discontinuities without mesh alignment or explicit interface tracking, while retaining the exponential p-convergence of high-order elements. A key feature of DFCM is its ability to treat both weak and strong discontinuities by decomposing cut cells into separate material regions without modifying the background mesh. These decomposed subcells interact through Nitsche-type flux terms that ensure mechanical compatibility. For weak discontinuities, DFCM operates directly on the original cells with different material properties and appropriate penalty scaling to maintain coercivity. The framework naturally supports nonuniform-p formulations, allowing neighboring cells to use different polynomial orders. Interface coupling is carried out in a shared trace space, enabling stable and accurate transfer of solution information. This provides a practical mechanism for local p-enrichment with minimal additional degrees of freedom. Benchmark studies show that DFCM reproduces classical FCM results for continuous problems and accurately captures jumps in displacement and traction across embedded interfaces. The method remains stable for large stiffness contrasts and curved, non-aligned geometries, and its modular structure makes it suitable for 3D, transient problems, evolving geometries, additive manufacturing, and multi-material lattice structures.