Pressure vessels are compressed gas storage tanks designed to hold gases or liquids at a pressure substantially different from the ambient pressure. They have a variety of applications in industry, including in oil refineries, nuclear reactors, gas reservoirs. Pressure vessels are containers used to handle fluids which are highly toxic, compressible and which work at high pressures. Pressure vessels have applications in variety of industries such as Oil and Gas, Petroleum, Beverage industries, chemical industries, power generation industries, food industry, etc. Failure of pressure vessels has adverse effects on the surrounding and the industry which can cause loss of life, property and damages. The design of pressure vessel depends on factors such as pressure, temperature, material selected, corrosion, loadings, and many other parameters depending on the applications. This report elaborates the work done in design of pressure vessels to reduce failures in the pressure vessels and study of the parameters such as operating pressure and temperature, developing stresses, modes of failure etc. which cause fatigue or permanent failure or stress concentration in the vessels. The use of Finite Element Methods and Analysis techniques that provide results on failure in pressure vessels are to be studied. The future scope and advancements in pressure vessel design with software is to be utilized for simulation before actual modelling takes place. A pressure vessel is a container having a pressure differential with respect to the atmosphere. The purpose of a pressure vessel is to store or process a high pressure high temperature fluid. The fluid can be toxic such as chemicals as well as nontoxic such as steam. The pressure vessel has to be designed according to the standard available codes such as ASME (American Society of Mechanical Engineers) Section VIII Division 1, EN/DIN (European) Code, IS (Indian Standard) Code. These codes have been designed by experimentation to obtain standards that fit to any application. Generally, these codes are designed considering the factor of safety between 3 to 4.5. The design of pressure vessel is based on parameters such as pressure, temperature, corrosion, material selection, etc., the study of such parameters helps in designing of the vessel. Variety of materials is available which are to be selected according to the application. The failure of pressure vessel can cause loss of life, property and damage to the system for which it is used.