The purpose of this paper is to give and prove the fundamental theorem of conformable fractional calculus which is given only in previous associated papers for differentiable functions, and then we scrutinize the existence and uniqueness of solutions to some semilinear Cauchy problems for nonlocal conformable fractional integrodifferential or differential equations which their nonlinear terms include fractional derivative or fractional integral. Two examples are investigated to elucidate the main results.