Highly energetic astrophysical phenomena like supermassive black hole binary (SMBHB) mergers are predicted to emit prodigious amounts of gravitational waves (GWs). An anticipated component of the gravitational waveform known as "memory" is permanent and non-oscillatory. For SMBHB mergers, the memory is created primarily during the most violent moments of the inspiral immediately preceding the final plunge and ring-down when the strongest gravitational fields are at work and the non-linearities of general relativity are most pronounced. The essentially time-domain nature of memory makes it forbiddingly difficult to detect with ground based GW detectors, leaving pulsar timing array (PTA) experiments as the most promising means by which it may be detected and studied. In this paper, we discuss how GW bursts with memory (BWMs) influence pulsar timing experiments and develop methods to assess how sensitive modern timing efforts are to such GW events. We discuss how PTA searches for BWMs can be used to constrain the rate of BWMs and how these constraints relate to information regarding the population of SMBHBs. ; Comment: Submitted to ApJ