The design of digital filters is a fundamental process in the context of digital signal processing. Being essential building blocks, digital filters are used in diverse areas ranging from medical imaging or audio and video processing, to highly sophisticated military applications (among many others). The purpose of this dissertation is to study the use of lp norms (for 2<p<infinity) as design criteria for digital filters, and to introduce a set of algorithms for the design of Finite (FIR) and Infinite (IIR) Impulse Response digital filters. The proposed algorithms are based on the idea of breaking the lp filter design problem into a sequence of approximations rather than solving the original lp problem directly. It is shown that one can efficiently design filters that arbitrarily approximate any desired lp solution (for 2<p<infinity) including the commonly used l infinity (or minimax) design problem. A method to design filters with different norms in different bands is presented (allowing the user for better control of the signal and noise behavior per band). Among the main contributions of this work is a method for the design of magnitude lp IIR filters is also presented together with its corresponding algorithm. Experimental results show that the algorithms in this work are robust and efficient, improving over traditional off-the-shelf optimization tools. The group of proposed algorithms form a flexible collection that offers robustness and efficiency for a wide variety of digital filter design applications.