Given a family \(\mu_{s,t}\) \((0\leq s<t)\) of p.m.s on \(\overline\mathbb{R}_ +=[0,+\infty]\) with the property \(\mu_{s,t}\circ\mu_{t,u}=\mu_{s,u}\) \((0\leq s<t<u)\), where \(\circ\) denotes a generalized convolution operation [cf. K. Urbanik. Stud. Math. 23, 217-245 (1964; Zbl 0171.395) and Colloq. Math. 55, No. 1, 147- 162 (1988; Zbl 0661.60027)], one can define transition probabilities \(P_{s,t}(x,B)\) for \(x\in\overline\mathbb{R}_ +\) and a Borel subset \(B\) of \(\overline\mathbb{R}_ +\) by \(P_{s,t}(x,B)=\delta_ x\circ\mu_{s,t}(B)\). A Markov process \(\{X_ t\}\) with transition probability \(P_{s,t}\) is called a generalized independent increments process \((\circ\)-i.i. process). If \(\mu_{s,t}=\mu_{0,t-s}\) \((s<t)\), the process is called \(\circ\)-Lévy process. It is proved that \(\{X_ t\}\) has many properties similar to that of ordinary processes with independent increments. In particular, every \(\circ\)-Lévy process is a strong Markov-Feller process. Other problems related to infinitesimal operators and self-similarity of \(\{X_ t\}\) are also investigated.