The global potential energy surface of C4H4 in the lowest triplet electronic state have been studied at the G2M(RCC,MP2) level. Of 28 distinct isomers the most stable are aromatic cyclobutene q3 ((3)A(1g),D-4h) and linear butyne (E-3,D-2d), and 66 transition states for various isomerization and dissociation pathways have been found. The information about the global PES is applied to describe the potential energy surfaces for the C(P-3(j)) + H2CCCH2, C(P-3(j)) + H3CCCH, and C-2((IIu)-I-3) + C2H4 reactions, recently studied experimentally in crossed molecular beams. The reaction of the carbon atom with allene is shown to occur by a barrierless addition of C to the C=C bond to yield the three-member ring structure tl and/or to the central carbon atom of allene to form the branching structure bl which isomerizes to tl with a low barrier. tl undergoes ring opening to cl with a barrier of 9.4 kcal/mol, and the latter emits a H atom to give the major reaction product n-C4H3 With an exit barrier of 2.2 kcal/mol. The minor reaction product, i-C4H3, is formed through a 1,2-H shift in cl leading to c3, followed by the hydrogen loss. The reaction of the carbon atom with methylacetylene starts with a barrierless C addition to the beta-c-atom of H3CCCH to form c6, to the alpha-C-atom to give b2 or b2', or to the acetylenic C=C bond to yield the cyclic isomers t4 or t4'. c6, b2, and b2' are metastable and would rapidly rearrange to the linear isomer c5 or to the cyclic t4 and then to t4' with low barriers. The latter ring opens to c5 with a barrier of 14.7 kcal/mol. At high collision energies, the major reaction product n-C4H3 is formed through the methyl hydrogen emission in c5 with an exit barrier of 5.8 kcal/mol, The second product i-C4H3 can be reached via a K-shift from c5 to c4 and subsequent hydrogen elimination, Cyclic isomers C4H3 p3 and pi can originate from t4 t4', and t5, The reaction of C-2((IIu)-I-3) with ethylene proceeds through a C-2 addition to a carbon atom of C2H4 to yield the chain isomer c2 with an entrance barrier of similar to 4 kcal/mol, c2 rearranges to linear cl in several steps via the four-member ring intermediate q1 or the three-membered ring structures t2 and tl. cl loses a H atom to yield n-C4H3 or undergoes a hydrogen migration to c3 followed by a H emission producing n- or i-C4H3 Similarly to C(P-3(j)) + allene, n-C4H3 is expected to be the major product of C2((IIu)-I-3) + C2H4, while i-C4H3 to give a minor contribution.