Long resident in Chicago and published in The New Yorker magazine, Aleksandar Hemon is one of the A-listers of contemporary Anglo-American letters. Born in Sarajevo, Bosnia-Herzegovina, he grew up and did military service in the old Yugoslavia, the multi-national state of the south Slavs that was held together ultimately by little more than socialist rhetoric and the cult of Titoism, and ended in a series of devastating civil wars.

Hemon always considered himself to be a Yugoslav patriot, as did everyone he knew in Sarajevo. “Our other identities, say, the ethnicity of any of us, were wholly irrelevant,” he writes early on in The Book of My Lives. At least that was how it seemed to him, before the “ethnic cleansing” began and his home city was besieged by Serbian troops in 1992.

Like Conrad and Nabokov, to whom he has inevitably been compared, Hemon found his own exalted voice by choosing to write in a language other than his mother tongue, which in his case was Serbo-Croat, or “Bosnian”. His first collection of stories, the acclaimed The Question of Bruno, was published in English in 2000, eight years after he came to the US on a one-month tourist visa, just as war was beginning to fragment his homeland. He was 27 and would never return to live in Sarajevo.

The Book of My Lives is a memoir constructed from the often wry and sardonic autobiographical essays Hemon has been publishing since 2000. It is less a coherent whole than a series of vignettes, episodes and reminiscences with overlapping themes and preoccupations. The predominant tone is one of ironic detachment, soft-edged by nostalgia.

Hemon looks back on his Yugoslav childhood, adolescence and early adulthood, recalling the rituals that sustained him, the food eaten at the family table and in the army, the television programmes watched, the clothes worn and the music listened to, the poetry written and discarded, the arguments and adventures he had with his sister, the girls he loved and the games of football he watched and played.

He remembers precisely but not always fondly. “There was nothing to do, and we were quickly running out of ways to do it,” he says of student life in Sarajevo.

Then his American life abruptly began, with all its complications and excitements. Hemon settled in Chicago and the exile’s feelings of unease and displacement became his defining subject. Telling stories and creating characters was the means by which he tried “to understand what was hard for me to understand”, which, for him, “has been nearly everything”.

In his early years in Chicago he experiences the “situation of immigration [as a] kind of self-othering”. He animates and interrogates the peculiarity of the exile’s condition – of what VS Naipaul called the enigma of arrival. He shows us how it feels to occupy the spaces in between different cultures, not feeling quite part of the new country and yet permanently estranged from the place you still call home.

Reading Hemon’s evocations of a Yugoslavia that exists now only in memory one is reminded of the essays of Joseph Roth in which he fondly recalls, from exile in Berlin, the dying days of Austria-Hungary, with its clamour of competing nationalities and ethnicities but also its bureaucratic homogeneity and ties that bound the old empire and its disparate peoples together.

The Book of My Lives closes with the harrowing account of the loss of one particular life, that of Hemon’s second child, his baby daughter Isabel. You are taken remorselessly through the initial symptoms, the diagnosis of a brain tumour, the gruelling treatment and surgery and then, finally, the baby’s death.

The essay is difficult to read. So deftly is it written and so profound is the author’s love for his daughter that you never stop hoping Isabel might live even as her plight becomes ever more desperate. It must have been extraordinarily painful to write. There are expressions of anger and dismay and a sense of numbed futility, but all are held in check by Hemon’s artistry and narrative control. He is a grieving father and he is a writer, and the writer mitigates and channels the father’s grief. “Memory narrativises itself,” he has said in various interviews.

“One of the most despicable religious fallacies is that suffering is ennobling,” Hemon writes towards the end of the book. “Isabel’s suffering and death did nothing for her, or us, or the world.” He and his wife learned no lessons: “Isabel’s indelible absence is now an organ in our bodies whose sole function is a continuous secretion of sorrow.”

It’s a sombre way to end an often amusing book which, for all its themes of loss and displacement, tells a story of success – of how one young man from Sarajevo found his place and his voice in America and evolved a language in which to make sense both of what he’d left behind and what he embraced in his new life.