The study of braid groups and their applications is a field which has attracted the interest of mathematicians and computer scientists alike. We begin with a review of the notion of a braid group. We then discuss known solutions to decision problems in braid groups. We then prove new results in braid group algorithmics. We offer a quick solution to the generalized word problem in braid groups, in the special case of cyclic subgroups. We illustrate this solution using a multitape Turing machine. We then turn to a discussion of decision problems in cyclic amalgamations of groups and solve the word problem for the cyclic amalgamation of two braid groups. We then turn to a more general study of the conjugacy problem in cyclic amalgamations. We revise and prove some theorems of Lipschutz and show their application to cyclic amalgamations of braid groups. We generalize this application to prove a new theorem regarding the conjugacy problem in cyclic amalgamations. We then discuss some application of braid groups, culminating in a section devoted to the discussion of braid group cryptography.