Hydraulic fracturing is a process in which fractures are generated in the rock by injection of highly pressurized fluid. Hydraulic fracturing technique together with horizontal drilling allowed to effectively increase oil and gas recovery from low permeable shale formations. The global behavior of a hydraulic fracture is strongly influenced by the processes occurring near the fracture tip, which are related to rock toughness, fluid viscosity and leak-off. The near tip region is modeled as a semi-infinite fracture. The governing equations include elasticity equation, lubrication equation and a propagation criterion. Analytical solutions can be found only for the particular cases of toughness, viscosity and leak-off dominated regimes of propagation. To find a general solution, we employ non-singular formulation to solve the problem numerically. To study the effect of fluid yield stress, the problem of a semi-infinite fracture driven by Herschel-Bulkley fluid is investigated. Numerical results demonstrate that the yield stress influences fracture width solution at larger distances from the tip. At the same time, the solution follows the behavior of a power-law fluid ahead of this zone. Analytical solution for the yield stress dominated regime is obtained and boundaries of its applicability are found. The near tip behavior of a hydraulic fracture can also be strongly affected by proppant - granular material which is mixed with fracturing fluid to prevent fracture from closing after the pressure is removed. Proppant can accumulate near the fracture tip due to settling, bridging, and/or dehydration of the slurry. To investigate the effect of proppant, the problem of a semi-infinite fracture with a localized proppant plug near the tip is analyzed for the case of Newtonian fluid. Fluid filtration through the proppant plug is modeled according to Darcy’s law. Boundaries of the proppant plug are determined by a particle-size dependent bridging criterion and total volume of particles. Proppant causes a noticeable pressure drop over the plug, which in turn leads to fracture widening behind proppant. The effect of proppant can be equivalently represented by a stress barrier solution without proppant. Expressions for magnitude and location of the stress jump are explicitly calculated. Results indicate that such a representation leads to a solution that agrees reasonably well with the numerical solution with proppant.