Research in the field of magnetic shape memory alloys (MSMAs) has increased in recent years due to their great interest in their potential applications in smart devices because the reversible deformations undergone. These applications range from actuators, sensors to a magnetic refrigerator and a micro-energy transducer. The well-known example of these alloys is the Heusler type, which has excellent properties, such as; the metamagnetic properties, the magnetocaloric effect (MCE) and the giant magnetoresistance effect (MR). The characterization of the different structures of this type of material is vital for the understanding of their macroscopic and microscopic behavior. The structures of these Heusler alloys are: a high temperature L21 and B2 austenite and L10, 10M and 14M martensite a low temperature. This paper provides a comprehensive review on the recent progress in the development of magnetostructural transformation and magnetocaloric effect, as well as the shape-memory effect induced by the magnetic field in Ni-Mn-X (X= In, Sn, Sb) and Ni-Co-Mn-Y (Y= In, Sn, Sb) Heusler-type MSMAs. The possible challenges and remaining issues are briefly discussed.