The theme of this text is "hard computational problems made accessible". It is a collection of essays on modern computational problems whose solutions are difficult, if not impossible, to achieve and offers solutions to important problems in modern scientific computation. The topics covered should be of broad interest to professionals working in the areas of chaos and fractals, complexity, prime numbers and encryption, wavelets, fast Fourier transforms and signal processing. The book should also be useful to students and researchers as it focuses on solutions to problems, rather than on problem posing. It also contains explanations of the importance and origins of hard problems whose explanations are usually difficult to find in modern literature. In addition, the text includes exhibitions of state-of-the-art algorithms for these difficult problems, as well as actual code for difficult algorithms. Enhancement files, program code and other data are available via the TELOS Web site, including updates to the algorithm library, sound and graphics files. All programs are coded in C and "Mathematica".